
PRIRCE 





GIFT or 

lirs. G. N, LeTTis 



t 







SHORT TABLE OF INTEGRALS 



BT 



B. O. PEIRCE 



HoLLis Professor of Mathematics and Natural Philosophy in 
Harvard University 



SECOND REVISED EDITION 



GINN AND COMPANY 

BOSTON • NEW YORK • CHICAGO • LONDON 
ATLANTA • DALLAS • COLUMBUS • SAN FRANCISCO 



COPYRICIIT, ISVd, 1910 

By GINN and COMPANY 



ALL EIGHTS RESERVED 
522.2 




1 



Since I cannot hope that these formulas are wholly free 
from misprints, I shall be grateful to any person who will 
call my attention to such errors as he may discover. 

B. O. PEIRCE, 
Harvard University, Cambridge. 



GIFT 



tH)t Stfjenatum jPrees 

I'.IW" AMI CllMl'ANY • Vlli). 
PKIETURS • liUSTuN • U.S.A. 



Q A- =s / o 
HA If ' 



TABLE OF INTEGRALS. 



-o-o>0<o°- 



I. FUNDAMENTAL FORMS. 

1. I adx = ax. 

2. \ ctf(x) dx = a ( f(x) dx. 

3. f— = logx. noga: = log(-x) + (2A:+l)7r/.] 
J x 

oc'^dx = ' when vi is different from — h 

m -\- 1 

5. Ce'^dx = e'. 

6. C a^ log adx = a", 

7 I _^^- — = tSLU-^x, or — ctn-^a;. 



. = sin~^cc, or — cos ^x 

dx 



/ dx 
X Va;^ — 1 



10. . , 

V2 X — X 



= sec-^cc, or — esc ^x. 
-1 

dx 



— = versin-^a;, or — coversin-^a;. 

-J'2x-x^ 



M623267 



FUNDAMENTAL FORMS. 

1. I COS xdx = sin x, or — coversin x. 

2. I ^va.xdx = — cos x, or versin x. 

3. I ctn ic c?x = log sin x. 

4. I tan xdx = — log cos x. 

5. I tan X sec a; (fx = sec 05. 

6. I sec^icc?ic = tancc. 

7. I csc^ajt^x = — ctnx. 

In the following formulas, u, v, w, and y represent any 
unctions of x : 

8. i (u-\-v + w + etc.) dx = i udx + i vdx + i wdx-\- etc. 

9 a. i udv ^= uv — i V du. 

_ /• c?y , f du ^ - ' 

9 6. \ u—-dx = uv — \ v-r- dx. 
J dx J dx 



.o.//(.)^=//Mf^ 



dx 



RATIONAL ALGEBRAIC FUNCTIONS. 



II. RATIONAL ALGEBRAIC FUNCTIONS. 



A. — Expressions Involving (a + hx). 

The substitution oi y ov z for x, where y = a -\- bx, 
z = (a + bx) / X, gives 

21. ("(a + bx)^dx = ijy^dy. 



22 
23 



. Cx{a + bxYdx = y,yf {y - «) dy- 



r x'^dx ^ _i_ r {y-(^Tdy . 

^ J Ca + Z^x)"* i»+V 



25 



(a + Z^x)"* i"+V 2/'" 



■J a;«(a4-te)'" «""+"- V 



Whence 

dx 1 

(Zx _ 

+ bxf " 

dx 1 



27. fy- 

J (a 



r dx ^ 1 

^^- J (a + ix)« 2 6(a + te)= 






6 RATIONAL ALGEBRAIC FUNCTIONS. 

r ^^ix _ 1 r 1 a ~| 

J {<i + te)' ^z-' L « + ^* 2 (a + bxy\ 

32. /;f^ = ^3 \i (* + ^^)' - 2a (a + hx) + a" log (a + bx)\ 



34, 



35 



/ dx _ 1 ■■ a + ^>x _ * 
a; (a + 6j:;) a x 

/dx _ 1 _ \_ , rz-t-^A-r 
xfa+^a;)^ a(a+bx) a^ 



(a+bxf a{a+bx) a^ x 

37. C (a+bxY {a'+b'x)'"dx = - — ■ — -y ( (a+bxy + ' (a'+b'xf 

J ^ ^ ^ ' {in-\-n-\-\) b \ 

— m(ab'-a'b) C (a-^-bx)" (a' + b'xy-hlxj 

r {a + bxydx 1 f (a + hTY + ^ 

' J (a' 



I _L /,' ,• \in — L 



{a' + b'x)'" (m — l){ab' - a'b) \{a' + b'x) 

r(a + bxYdx 
+ (m — n — Z)b I -r-; z-^: r 

^ 1 / (a-\-bxY ' 

~ (m — n-l)b'\(a' + b'xy"-^ 



+ n 



^ ^J {a' + b'x)'" J 

_ i f (a + bx)" r (a + bx)''^^dx \ 

* /'—^^^— =--'- + - log ^^i-^ . 
J x\a + bx) ax a"^ " x 



39 



RATIONAL ALGEBRAIC FUNCTIONS 

dx 1 , ft' + Vx 

(a + bx) {a' + h'x) ~~ ab' -a'b' °^ a-\- bx 

dx 



40./ 



(a + bxy {a' + b'x)"" 

= I ( 1 

(m - 1) (ab' - a'b) \{a + bx)"-^ (a' + b'x)"'-' 

dx ^ 



41 



- (m + n-2) bC 

f 



(a + bxy (a' + b'xy"-\ 

xdx 
(a + bx) (a' + b'x) 



42, j 



43 



(a + bx)\a' + b'x) 

= _J— f^- + — ^— log ^^^±^\ 

ab' — a'b \a + bx ab' — a'b a + bx J 

r xdx 



(CI + bxf (a' + b'x) 



— a a' . a' + b'x 

TTTz log ■ 



b (ab' - a'b) (a + bx) (ab' - a'by ^ a + bx 

x'^dx a^ 

(a + bxy (a' + b'x) ~~ b' (ab' - a'b) (a + bx) 

1 r«'% / r . 7-r N , a(ab'-2a'b) ^ . , . .1 
— ^ [y log (a' + b'x) + -^—j, '- log (a + bx)j 

/I n 2+i 



"^ («&' - 



.^ C dx n , ,-. ^~ 



(a + OXJn ^ ^ 



8 RATIONAL ALGEBRAIC FUNCTIONS. 

B. — Expressions Involving (a + 5x"). 

r dx 1, _iX 1 . _, X 

47. I -T", — -„ = -tan ^-=-sin i— ===• 

.^ C dx It c + x C dx 1 . .-r — c * 

48. I -^ 5 = — log > \ —^ i = 77-log — — -• 

J c^ — x^ 2 c c — xjx^ — cr 2 c x-\-g 

50. I — r^-i = — 7=log--= 7=j if a>0, b<0. 

J a + bx^ 2V^^ V^-a;V-6 

/* <?a; _ a; 1 /* c?x 

J (a + bxy ~ 2a(a + bx') 2aJ a + bx^' 

r dx _ 1 ^ _i_ ^ ^^ ~ 1 r rfa 

■ J (a + Z/x2)"*+i - 2 7/ia (a + bx^)"" 2 ma J {a + bx^)"^ 

" ./ a; ((X + 6x^) 2 a a + bx'^ 

/x^dx _x a r dx 
a + bx^ b bJ a + bx^ 



56 



r dx _ _ j_ _ ^ r_^_ 

'J cc^ (a + ix'^) ax a J a + 6x^ 

, a;^c?aj — x . 1 /* r/.r 

58. 



r x^dx _ — X '^ C 



(a -I- bxy'' + ^ 2mb(a + bx^"" 2 mbJ (a + bx^)'» 

r dx -^ c ^^ ^ r— -^^— ■ 

J x\a + ^<x2)'» + i ~ a J x'^{a + ix^)*" a J (a + 6x')'"+*" 

• p dx 1 



y^-s-"--©^ /i^.-i-."-©- 



RATIONAL ALGEBRAIC FUNCTIONS. 



r« 


-ii 


>« 


-«) 


(U 


<o 


Si 


u 


a> 


<o 


rSli 


rS, 


^ 


^ 


'- 


•^ 


1 
( 


1 


1 

1 


1 


r-* 


1 


•'« 


a 


c« 


05 


-t-3 


-1-3 


^ 


^ 


H 


h 


H 


h 










<ii e 



H 



8 
+ 






CO 

CO 



s 
Ji 



8 
+ 



8 
I 



+ 



« i<i 



^ 



s 



+ 



-§ 



8 






-i 



a 
+ 



8 



o 


bC 
O 

1— 1 




^ 


^ 


iHl 


Ho. 
1 


1 1 


II 


11 




e 




-^ 




^ 


'^ ^ 


tH 


-« 




+ 




CO 




CO 












II 


II 




^ 


-i 


w 


-§ 


-i 


H 

rJa 

+ 

e 


5^ 


+ 


-§ 


+ 


s 


+ 





I e 



+ 

a. 
+ 




I 
g 



CD 



CD 



CO 



CO 
CO 



10 RATIONAL ALGEBRAIC FUNCTIONS. 

C. — Expressions Involving (a -{- bx + cx^). 
Let X = a + bx -\- cx^ and y = 4 ac — b"^, then 

„„ rdx 2 ^ .2cx + b 2 ^ , .2cx-\-b 

67. I ^ = —7= tan-^ ■ y^- , or — —= ■ tanli-^ — 

J A -\/q -yjq V— q V — q 

„_ Cdx 1 - 2cx + b — V— a 

68. I — = — =log 7^' when 7 <0. 



V— 9' 2cx + b + V— 2- 



rdx 2cx + b 2e rdx 



6 c" /^(Za; 

Z' 



r dx _ 2cx-{-b 2{2n-l)c C dx 

-^ Cxdx 1 - ^^ & /^(/a; 

72 I = — lof X I 

' • J X 2c ° 2cJ X 

rxdx _ bx-\-2a b Cdx ^ 

J "X^ ~ Jx ^J x' 

r xdx _ 2 a + bx b (2 n — V) r dx 
■J A'" + i~~ nqX'' nq J X"' 

„_ rx^ - a; i T ^ b^ — 2ac rdx 

^«- J 5=5''-" = ^ ^ix + tJ x' 

r a;'" dx x"'-^ n — ??^ + 1 ^> /* a:"'-^rfa; 

Jx" + i~ (2 71 - 7M + 1) c.Y" 2?i-m + l"cJ X'" + i 

m — 1 a rx"'~^dx 

"*" 2 M - w + 1 ' c J X'' + i ' 



*dx 
X 



RATIONAL ALGEBRAIC FUNCTIONS. 11 

^„ rdx 1 , x^ b rt 

r dx __b_. ^ _ JL , /^i!_ _ i\ C— 

J x^X'^2 a" ^^ x" ax'^\2a'' aJJx' 

an C ^^ — 1 n + m — 1 h f* dx 

J aj'^X^+i ~ ~ (?» — 1) ax"'-'X" " m — 1 a J a;"'-iA'«- 

2n + m — \ c r dx 
m — 1 aJ x"'-2X» + i 

r_^ ^^ ^ C^^ 1 r dx 

J ^^X'~2a(r^-l).Y«-l 2 a J X» "*" a J xX«-i' 






/" (a' + ^>'cc)(Zx _ j; 2a'c-bb' rdx 

^^•J X» ~ 2(w-l)cX"-''^ 2c J X»' 

86. r (a' + i'a:)'" X« dx = — ^ — ^— ( ^-'C^^' + b'x)"'-'X"+ 

J ^ ' {in -\r 2 n At V) c\ ^ ' 

+ (m + n){2 a'c - 6^>') f(a' + 6'a;)'"-iX"c?a; 

- (m - l)(a6'=' - aW + ca'^) ("(a' + b'xy'-^X«dx\ 



12 RATIONAL ALGEBRAIC FUNCTIONS. 

r (a' + b'x)'"dx _ 1 f (b + 2 ex) (a' + b'xf 
J J:» ~q{n-l)\ .Y"-i 

-2(m-2n + 3)cJ ^ — -j;^^ 

+ m(2 a'c - bb')J ^ j;;^! J 

_ 1 f b' (a' 4- ^''x)"-^ 

~ {m-2n + l)c\ X"-"^ 

+ (m-n) (2 a'c - i&') J ^^ ji 

- (m - 1) (ab'' - aW + ca'') f ^^-^^^^^^~^\ 

J (ct' + i'x)"' 

1 / - VX^ 



b"{m-l)\(a' + b'x)'—^ 

X'^-'^dx 



+ n{bb'-2a'c)^— 



(a' + b'x)"'-'^ 
. „ C X—'dx \ 

1 / 4- ^-'X" 






RATIONAL ALGEBRAIC FUNCTIONS. 13 

89 C—^^— 

J (a' + ^»'x)"'X» 

b' 



n v(«' 



(m -1) (ab'^ - aW + ca''') \{a' + ft'a;)"— ^X"-' 
1 / 6' 



2 {ab'^ - aW + ca'^) \{n - 1) (a' + ^»'a;)"'-U'"-» 

+ (2 a'c - bV) (—TT-^. TF- 

(m + 2n-^)b'^ r dx \ 



If aJ'2 - aW + ca'^ = 0, 
dx 



J Ca' 



(a' + b'xyX" 

1 (- 

(bb' -2a'c)\(a' 



(m + 71-1) (bb' - 2 a'c) \(a' + b'xyX''-^ 

+ (m + 2n-2)c f—r-nr^ — pf-Y 
'^ ^ J (a' + b'x)"'-^X"J 



D. — Eational Fractions. 

Every proper fraction can be represented by the general 
form : 

f(x) ^ g,x"-' + gr,x"-^ + g.x'^-' + . . ■ + ^^ 
F(x) x" + kix"-^ + k^x"-- + • • • + A;„ 

If a, b, c, etc., are the roots of the equation F(x) = 0, so 
that 

F(x) = (x- a)P {x — by (x - cy • • -, 



14 RATIONAL ALGEBKAIC FUNCTIONS. 



then 

F{x) {x-ay {x-a)P-^ (x-ay-^^ x-a 

(x- by {x- by-' '^ (x- by--' '^ x -b 

1 ^1 ^_ ^2 , ^3 |_ . . . 1 ^'r 



(X — Cy (X — 6')*' ^ (X — C)''~^ iC — C 

I ••• ••• ••• ••• ■•• 

where the numerators of the separate fractions may be deter- 
mined by the equations 



•) 



_, , , f(x) (x -a)' , , ^ fix) (x -by , 



If a, J, c, etc., are single roots, then j? = gr =?•=••• = 1, 
and 

i''(a;) cc — a x — b x — c 

The simpler fractions, into which the original fraction is 
thus divided, may be integrated by means of the formulas : 

. hdx _ r h d (inx -{- ti) _ h 



r hdx _ r 

' J (mx + 71)' J 



(mx + ny J m(mx + n)' m(l — l) (mx + ii)'-' 

and I ■ — = — log (mx -f n). 

J mx + n ni ' 



RATIONAL ALGEBRAIC FUNCTIONS. 15 

If any of the roots of the equation f{pc) = are imaginary, 
the parts of the integral which arise from conjugate roots 
can be combined, and the integral brought into a real form. 
The following formula, in which i = V — 1, is often useful in 
combining logarithms of conjugate complex quantities : 

log {x ± yi) = 1 log {x" + if) ± i tan-^ ^- 

The identities given below are sometimes convenient : 

1 _ 1 r h _ b' "I 

{a + bx'') {a' + b'x^) ~ a'b - ab' ' \_a + bx'' a' + b'x'^J 
7)1 + nx 1 



{k + Ix) {a + bx + cx^) al^ + chP' - bkl 

Vl(ml — nk) c(nk — '>nl)x + (aln + ckm — blfn)~\ 
\_ k -{- Ix a -{- bx -{- cx"^ J 



I + mx^ 



{a + te") (a' + i'^") a'b 



1 r^^ — am a'm — b'I~\ 

— ab'' \_a + bx" a' + b'x" J " 



1 ^ +JL^ + J^, 



(x + a) {x + b) (x + c) X + a x + b x -\- c 
where 

B = - —Z TJ C = 



{a — b){a — c) {b — c){b- a) (c - a) (c — b) 



(x + a)(x + b)(x + c)(x + ;/) x + a x + b x + c x + g' 
where 

^1 = T, : TT T ' ^ = Z 7T~/ 7T7 7^ ' ^tc. 

{b — a){c-a){g — a) {a - b){c — b){g — b) 



16 IRRATIONAL ALGEBRAIC FUNCTIONS. 



III. IRRATIONAL ALGEBRAIC FUNCTIONS. 



A. — Expressions Involving Va + hx. 

The substitution of a new variable of integration, 
y = Va + bx, gives 

2 



2 

^a^bxdx = — V(a + bxf 



«« C I T- , 2 (2 a - 3 ^>a;) V(a + bxf 

92. J xVa + bxdx = ^ ^g^^a ^" 



«„ r , / — T^- , 2 (8 a^ - 12 abx + 15 b^x^) V(a + Z»a;)« 
93. J xWa + Z-xc?x = -^ ^^^, '- ^ 



94. I dx = 2Va + Z^x + a I — ■ 

98./- 



(/a; _2^a + bx 
i + bx ^ 

96. I -7= = ^^r7-„ Va + bx. 



Va + bx 

/xdx 2(2 a — bx) I 
■ I = ^ Q7.2 Va + 
v./ 4- /ax 3 ^>2 



„„ f a;2t/a; 2 (8 a^ _ 4 a^,a; + 3 ^»V) / — 

97. I , = ^ .^ ,„ ^ Va + ^aj. 



00 r ^^ 1 , / VaTTx — Va . „ ^ „ 
98. I — ■ = — ^ log ( , — ], for a > 

99 



a; Va + bx Va \ Va + ^a; + Va/ 

/c?a; 2 ^ , (a+^ —2 , , ia+bx 
— ■ = . — tan-i \^ , or —-= • tanh-^ \ 



IRKATIONAL ALGEBRAIC FUNCTIONS. 17 

dx Va + hx ^ C ^^ 



, «« r dx ■\Ja-\-hx b r d 
100. , = 77- I 7= 

•^ x^s/a + bx «^ ^^'^ xy/a 



+ bx 

2±n 



4± n 2 ±n 

102. J .(. + fa)*.& = p [_L__i L__LJ. 

r a;^c?a; _ 2a;"'Vtt + to 2 ma r x'^-^dx 

' -^ Va + bx (2 w + 1) i (2 ?/i + 1) bJ Va + ^ 

/* (/a; _ Va + ^>x (2?^-3)& T dx 

' '^ x"V^i + bx~ (n-l)ax"-^ {2n-2)aJ x"-Wi 

105. Jfc±M*: = , J(, + fa)"-i-=rf, + a/(^ 

n dx 1 /* (/a: b C dx 

106. 



n — 2 



da;. 



, /^ dx _ 1 C ___dx b^C 

x{a + bxy^ x(a + ia;) 2 (a + te)2 

107. ^ fix, V^^x)dx = '^fff^lL:-^, :J\z—-'dz, 
where z" = a -\- bx. 

m + n 

/, , , !^ , n(a-}-bx) « 
(a + bx) n dx = - \,,\ 

/m p 

f(x, (a + bx)", (a + bx^, • • ')dx 

where f = a + bx, and s is the least common multiple of n, 
q, etc. 



18 



IRRATIONAL ALGEBRAIC FUNCTIONS. 



B. — Expressions Involving Both Va + hx and Va' + b'x. 
Let u — a + bx, v = a' + b'x, and k = ab' — a'b, then 

dx 



k + 2bv r- ^' 
dx = — . .,, Vmv 



110. r^^ 

/ -^ vdx _ 1 / — h_ C 
V^ & 2bJ ^uv 

xdx V?<v ab' + a'b C dx 



/ 



111. 

112. / 
113./ 



4 bV ' "" 8 66'c; v^ 
^c dx 1 / — h r dx 



\ uv 
dx 



bb' 2 iZ- 

2 



?^/: 



'wy 



log(V^»^;'w + ^»V?;) 



2 ^ . I- b'u 2 ^ , 

= — , tan~ '■ \\—, J or — == tanh" 




1 . , 2 bb'x + a'b + ab' 
I sm- ^ 



114. f- 



1 , b'V^-^/kb' 
lop- 



116, 



dx 

V^ ~ Vkb' ^^^ 6' vw + Vkb" " ^^kb 

dx 2 V« 



2 , ^;'V'w 
tan~^ — ^ 



-^-kb' 



/dx 
v\uv 



kV^ 



116. /.."vr,.. = (2;;^,(2 .— v;; + ;./^). 

rV^^ 1 / 2Vw r dx \ 

(7/1 - 1) b' \ i;'"- 1 ^ ^ "J „m-l ^y 



v"'-'^dx^ 



IRRATIONAL ALGEBRAIC FUNCTIONS. 19 

/dx 1 / Vm , , ,. , r d-^ \ 



120. fv^u^-idx = — —. — — r-Y 2 y'^ + iM'-i 

J (2 7H -j- 2n + 1)0 \ 

+ (2 w - 1) k Cv^'u^-^dxy 



1^ 

(2n-l)k 



121. Cv'"u-<'' + i^dx = — — ^ ,, , ( 2 y'« + i?t-("-J) 



(2 m - 2 w + 3) b' Cv"'u-^"-^^dx j 



— v"'u-^"-^^ 



(2n - 1)^'' 
+ mb' Cv'"-'^u-'^"-i''dx \ 



122. Cv-'^u^'^-i^dx = tA T-rA 2 ti"-^-^'"-'^ 

J (2 m — 2 ?i — 1)6' \ 

+ {2n- l)kCu''-^^v-"'dxj 



— u^ — ip—^™ — ^^ 



(m- 1)^''' 
+ (*^ - i)^ r«"-it'-('"-')rfa; j. 

123. ry-'"%-("+»<^a; = — -— -( 2 7-('»-i>w-("-« 

J (2 w - 1) ^ V 

+ (2 m + 2 ?i - 3)b'Cv-"'ir <"-«c?a; j. 



20 IRRATIONAL ALGEBRAIC FUNCTIONS. 



C. — Expressions Involving ^x^^a? ani> ^a? — x^. 
24. f Vx2 ±a?dx = \ [x Va;2 ^ci'^a? log {x + Vx^ ± a^)].* 






X 



dx . , cc .a; 

— =r sip-t-, or— cos~^-' 



28. 



29 



/c?a; 1 .a 1 , x 
; = - cos~^ -1 or - sec ^-■ 
x^x^ — a? a X a a 



r dx __i, A, + v.-±.^ y 

-^ xVa-±a;2 « \ a; / 

on C ^«^ =*= ^^ 7 /~^ ; 1 «. + Vft^ ± X^ 

i*"- I rfx = Va"* dr x'' — a lo£? ■ 

^ X - X 



r V.r- - a^ /— : a 

I — — dx = Vx^ — a^ — a cos -• 

»^ X X 



32 

^ V 

„„ /* xt^x /-^ 

33. 1 , = Vx2 - a". 

*^ Vx^ — o? 



34 



fxVx^ia^^^x = iV(x2±a2)S. 
35. rxVa--x2(^x = - ^ V(a2 _ a:2)3. 



IRRATIONAL ALGEBRAIC FUNCTIONS. 



21 



136. C>/{x^±aydx 



2 z 



dx 



dx 



:X 



= ^\ x-\/(x^±d 

137. f-\{a^-x^ 

138. f 

139. /- 

140. /- 
141 I ~ 

* J -».//^2 _ o,.2\3 -v/«2 _ ^a 



»')]■ 



V(a;2 zh ay a^ Vx^i^ 

dJx X 

iCf^a; — 1 

xc?a; 1 



142. fa; V(^2^^(ix = ^y/(x'±ay. 

143. Cx^J^F^^^^dx = - |V(^2_ 3.2)6; 

144. faj^ Vx=^ ± a^cZx 



= ^V(x2±a^«rF|x-N/^^±^^-|log(a;+Vac*-ba?) 



8 



a;'e?x 



145. Cx^y/^^ 

log 2 = sinh - J (^^) = cosh - 1 (^^7^) ; tanh- 1 2 = - i • tan- \ziy 

* (See Note ou pages 20-21.) 



22 IRRATIONAL ALGEBRAIC FUNCTIONS. 



/Va^~±x^ dx ->Ja^ ± x^ IT dx 

X Lx -^ a-Va^'zhx^ 

147, fx-Va^ + x^ (^x = (± 1 x2 - ^2^ a2) V(«2 i x^)^ 

/^ fix Va^ ± x^ 1 /^ <^x 

148 



xWa-±x- ^^^ ^ la-J X 



-^a? ± x" 



C ffx Vx^ — a^ , 1 , /x\ 

J x'^xJ' — o? 2a-x2 2^3 \^ay 



160. I , = o Vx'^ ± a^ ^ - log (x + Vx^ ± a^). 



151. I = — o Vct^ — X- + 77 sm !-• 



, ^„ . c?x Vx^io^ 

152. 



J /• rfx 

^ x^Vx^ zh a.' 



2 a^x 



183. f- ■'^ "'""-" 



x' 



154 



^ = + log (x + Vx» ± a^. 



X- X a 



156. f / = ,--^ 4-log(x+V^^T7^. 



X^(?X X . , 35 

, - — = ■ , — sm-i-' 



• (See Note on pages 20-«l.) 



IRRATIONAL ALGEBRAIC FUNCTIONS. 23 

air \ du 



158. Cfj^^ = ,jCf( 



g'^ — cu^ j (^^ — cu^^ ' 



where u — 



159. I , - = - \ f( 1 du, where u^ = a -^ cx^. 

J Va + ex'' cJ ' \ c J 



D. — Expressions Involving Va -\- bx + cx^. 
Let X = a -\- bx + cx^, q = 4: ac — b^, and k = In order 

q 

to rationalize the functio n f(x, Vft + bx + cx^) we may put 
Va + ^x + ex' = V± c V^ + Bx ± x^, according as c is posi- 
tive or negative, and then substitute for x a new variable z, 
such that 



z = ^A + Bx + a;2 ± a-, if c > 0. 

V^ + Bx — X' — vCi a ^ 

z = ■} if c < and >• 0. 

X — c 



= \- -f where a and /? are the roots of the equation 

^ a — X 



a 



A -\- Bx - x^ = 0, if c < and < 



— c 



By rationalization, or by the aid of reduction formulas, may 
be obtained the values of the following integrals : 

160. C^ = ^-logfVX-\-xVc + -^\iic>0. 
^ -\fx Vc \ 2Vcy 

/dx —1 /2c.r + i>\ 1 . , ,/2c.>' + h\ 



24 IRRATIONAL ALGEBRAIC FUNCTIONS. 

' -^ X^fX q^/X 

r dx _ 2(2cx + b)Vx 2k {71- 1) r dx 
^^" J X" VX ~ (2 n - 1) ^A'« "•" 2 w - 1 J X«-i Vx' 

67. Cx^VXdx 

^ {2cx + b)^X f hX , 15\ 5 r dx 

12c \ '^ 4k Sk'J Wk^J VX 

68. rx-vx..^ (^-+^)f:^^ + ,;"V, r^. 

J 4(7i + l)c 2(% + l)A;J Vx 

/xdx^_ Vx b r dx 
VX~ c '2 c J Vx 



dx 

Vx" 



69 



/* a;c?cc _ 2 (^ij; + 2 «) 

■ J xVx ~ ^Vx 

/ xdx Vx A r ^^ 
X"VX~ (2ri- l)oX«~2"J J^VX' 

"• J vf = (ro - jp) ^ + -87- J vx • 

' J xVx c^Vx c J Vx' 



IRRATIONAL ALGEBRAIC FUNCTIONS. 



25 



174 



Vn 



x^dx 



x^Vx 



175 



{2b''-4:ac)x + 2ab 4 ac + (2 n - 3 ) 5' P dx 
{2n-l)cqX'^-^Vx (2n-l)cq J X^-^Vx 

x^dx 



■f 



Vx 



'x^ 5bx 5b^ 2a' 
3 c 12 c2 "^ 8 c3 3 c2 



X + 



'3 ab 5 b^ 
4 c^ 16 & 



)f 



dx 

Vx' 



176. fxVXdx = ^^^^~~ f-yXdx. 
J 3 c 2 cJ 

177. fxXVxdx = ^V-^ - ^ fxVx dx. 
J 5 c 2 cJ 

xX^dx X'^VX 



P 



S 



X^dx 



Vx (2*i + l)c 2cJ Vx 
179. Cx^VXdx =(x-^}j 

180 



5 b\ X^X 5P-4ac 



4:C 



16 c2 



/Vx 



c?a;. 



r x'^X'^dx _ xX'^Vx _ (2w + 3)^' r xX'^dx 
' J Vx ~2(?i4-l)c 4(7i4-l)cJ Vx 



^— r 

2 (71 + 1) c J 



181. Cx^Vxdx = (x^- 



X^lx 

(n + l)cJ Vx" 

8 c 48 c' 3cy 5 c 
'3ab 



+ 1 



8c2 



32cvJ 



VXrf; 



'JCt 



-^ xVX Va \ a; 2Va/ 



26 



IRRATIONAL ALGEBRAIC FUNCTIONS. 



183. I — = — ^sm-M -.= ] , or — = smh ^ ^• 



184. f ^ 

dx 



2VX 

— , ' II a = 0. 



ox 



185. 



d 

xX" 



Vx 



Vx 



(2 n - 1) aX 



1 r dx b_r 

" aJ -rX^'-'^^fx 2aJ 



dx 



186 



■/ 



dx 



Vx 



a;'VX 






dx 



x«Vx 



«« 2aJ a;VX 



188 



»^ X 



X 



189. fj^^iKl^ 



Vx (2 ?i - 1) Vx 

Vx 



'/ 



X"-''dx . ^> rx^'-^dx 



X 



Vx 



iC^ 



a? 



+ « I ^ + 

dx 

.xVX -' Vx 



U-- 



VJ 



2-' xVx ^ 



dx 



190 



Vx' 



191, 



r x'^dx _ 1 r x'^-^d,x _ h_ C x-^-^dx a Cx'^' 
' ^ x»Vx c J X'^-WX cJ A'«VX cJ x» 



f 



■-r"'X"6ga; _ x"'~'^X"Vx _ (2n + 2m-l)h P x^^-'^X^dx 
Vx ~ {2n + m)c 2c(2n + m) J VX 



{vi -\)a r x'^-^X'^dx 
{2n + 7n)cJ Vx 



192, 



r dx 



Vx 



(2n+27n-3)l> 



(w — 1 ) ax'" - ^X " 2(t{m~l) 

?i + m — 
{in — 1) « 



'> r dx_ 



'Vx 



(2 ?^ + m - 2) c Z' 



rf.r 



x 



m-2^Y«VX' 



IRRATIONAL ALGEBRAIC FUNCTIONS. 27 

■ 

r X"dx _ _ x"-'Vx (2?i-i)^> r X"-'(ir 

(2 7^ - 1) r X''~^dx 
m. - 1 J ^-'"--'VX 

194. r/(a-, V(a; - a) (x - ^')) fZx 

/\hu^ — a u (b — a) [ u du 

where u^(x —b) = x — a. 

E. — Expressions Involving Products o f Powers of 

(a' + b'x) AND Va + bx + cx\ 

Let X = a + bx + cx^, v = a' + b'x, q = 4:ac — b^, 
/3=-bb' -2 a'c, k = ab'^ - a'bb' + c«'^ then 



195. I — — = —j=log "^ — 



yVX VA; 



tan"^ 



V- k 2V^-kX 

= sin~^ 7=="' iiA;p£0. 

V— ^ b'v^—q 

„ r dx 2 b' Vx . „ , „ 

196. — p= = 7. ' if ^ = : 

^ v^X 



(So 



r dx _ ,_ ■> /^ + 1 , 



/- _^ ^ _ &'Vx _ _^ r dx 

■J r^Vx" ^'^ 2aJ vVx 

■Jy^VX 3/3i;^ 3/3J ^;VX 



28 IRRATIONAL ALGEBRAIC FUNCTIONS. 

200 f ^dx ^ 2(2k+Jv) 

««, rvdx b'^^ B r dx 

201. -7= = TT I -1^' 

202. Cv^Xdx = ^-^^^-i~^Vxdx. 

r vdx __ _ b'Vx _ ^ r dx 
J X" Vr " ~ (2 .. - 1) cX» 2 J A'« VX ■ 

r^^x^ _ &'x»Vx _ _^ rxv^ 

J VX ~ {2n + l)c~ 2cJ VX ' 

r (Za: _ _ bWx _ (2m-3)f3 r dx 
■ J ^'«VX (m — 1) A-y™-' 2(m — 1)A-J ^,m- iVX 

_ (m-2)c p dx _ ^ ifyt-z£0. 
(m — l)A-c/ ym-aVx 



206 



1/ y; 



,y-«Vx (2 m -1)^^;" 



_ 2(m-l)c f _^^_ ■ . , _ rt 

{2m-l)(^J ,.»-iVx' 

r Vx^x _ _ ^>'XVX _ (2m-5)/3 T Vx^ 
J t;-" " (m-l)A;v"'-i 2 (m - 1) ^- J t-""-! 

(m - 4) c rVxdx 
~ (m- l)kJ v"'-^ 

~(w-l)^*'2(^ *'"'-■ ^^J i;'"-'Vx"^ ''J i;"'-2Vx^ 
_ 1 /_ &'VX _ r dx _ J r rfa; \ 

~ (m - 2)6'='(^ v—' V v'-Vx ^^J v-'-'VxJ 



IRRATIONAL ALGEBRAIC FUNCTIONS. 29 

208. (v'-<Xdx=- — \-—-{vv^-^X^'x 
J {in-\- L)c\ 

- {m+ ^) ^C v'"-'^^ dx- (m-l)kC v"'-'-Vx dx\ 

^^« f* dx 

209. 1= 

1 / h'Vx , , , ^^^ c dx 

+ (m + 2n-2)c f ^^ ^ Y if A;5z£0. 

■ J i'- .Y« VX ~ {2m + 2n-l)li\ v"' X» 

+ (m + 2?i-l)c f ^^-T=Y if A; = 0. 

* r X"dT 

211. I ^L^^, 



1= 



v^'-WxJ 

+ (2n-l)kf- 
,, ^ r X''-^dx\ 






A'»-ic?a; 



1 / &'X"-Vx ..^f x»- 



30 IRRATIONAL ALGEBRAIC FUNCTIONS. 



213 



Vx {m-\-2n)c\ 

v"" (Zx 1 fVv^-^ Vx 



'v^'-^X'^dx' 



/V" 
Xn 



Vx (m — 2 ?i) c \ A'" 

^ '^^J X«VX ^ ^ -^ X"VaJ 






+ 



{x + a) (x + Z*) VX (S - a){x + a) Vx (a — b)(x + b)^X 

1 



Va + bx -\- cx^ ± Va' + b'x + r'j:- 



_ Va + 6a; + (?a;^ ip Va' + b'x + c'. 
a — a' + ((^ — ^') a; + (c — c') x^ 

Vx Vx Vx 



a;2 



(x + a)(a; + 6) (^' — a)(a; + a) (a — Z») (cc + J) 

(a- + a)VX _^rf^^ (a-Z^)Vx 
X + 6 x + b 



C lax' + Z* _ . „. . . 

c/ \ ' -2 j_ i i^^ IS ^^ elliptic integral. 

/cc Va + bx^ , 1 r , 
; dx = — I Wab' — a'b + by^ ■ dy, 
Va' + b'x"" bWb'^ ^ ^ ^> 



' x Va + ^a;'-^ 
where y^ = a' + b'x\ 



MISCELLANEOUS ALGEBRAIC EXPRESSIONS. 31 



IV. MISCELLANEOUS ALGEBRAIC EXPRESSIONS. 



V2 ax — x^ •dx = — ^ — ^sr2ax — x^ -[- — sin"^ 



a 



«,^ r dx . . X 1 /-, ^\ 
215. I — - = versm"^ - = cos~^( 1 ) 

^ V2 ax - x2 ct \ */ 



^-« . a;"c?a; a,w-iV2acc — x^ 

216. 



V2aa;- 



^2 71 



a (I -2 n) C x^'-'^dx 



/ 



n J V2 aa; - x^ 

/ 



217 

*^ a;''V2 ax — aj^ 

+ 



dx a/2 ax — x^ 



x»V2ax-x2 a{\-2n)x- 

n — 1 C dx 



1) a J rpU—l 



(2 w - 1) aJ a;«-i V2 ax - x^ 



//- 5 , x"-iV(2ax — x^)" 
X" V2 ax -x^-dx= ^^-— r ^ 
71 + 2 

_^ (2.^ + l)a r„_,V2^^3T.., 
w + 2 J 



r V2 ax - x^ • 6?x _ V(2 ax - x'')^ 



71 — o /' V2 ax — x^- c?x 



— 3 /' V2ax -c 
(2 71 - 3) a J x»^ 

220. I — , = — sec-^ ( - )• 



32 MISCELLANEOUS ALGEBRAIC EXPRESSIONS. 



221 



•/ 



dx 1 , Va^ + x^ — a 

, = — log , 

a;Vx"-|-a2 an Va^ + a;" + a 



222 



223 



/x?dx 



%■ sin~l / — 



Sin 



CC"' 



a 



/ 



• (Za; 



(a + ^'a;2)Vx hV"^ 



log I 



'a; + 8' + V28^' 



+ tan-M 1 + 



'2x 



-tan-M 1 



\ 



'2x 



where iS* = a. 



c^.dx 1 r^ ,/^ , V2x\ ^ y. V2.t\ 



a + 6ic^ 



— log ( , — ) )■ , where hV' ^ f- 



V a + bx? 



225. 



^ -dx 2 V« 



/ x^ -dx _ 2Va; a T 



c?a; 



(a + hx?) Vx 



226 



c?a; 



Vac 



+ 



r 



dx 



227. 



■-^ (a + ^-x2)=2V^ 2a(a + Z»a;2) ' 4 a J (« +7»a;2) V^ 
/^ ^sfx-dx x^ . 1 r ^Jx-dx 



+ 



(a 4- ^'a;'-')^ 2. a{a -\- hx^ 4aJ (a + bx^) 



iaJ (a 



If tti, ttj? «8> etc., are the roots of the equation 

Pffic^ +^ia;"-i +x>2^''~'^ + • • • +i9„ = 0. 
the integrand in the expression 



/ 



{P(fic" + piX"~^ + • • • +p„) Va + 6x + cx^ 



MISCELLANEOUS ALGEBRAIC EXPRESSIONS. 33 

where m<in, may be expressed as the sum of a number of 

partial fractions of the form , ^==i and these 

{x — af?f^ a -\-hx-\r cx^ 

can be integrated by the aid of equations given above. Thus, 



228. f (P- + 9)dx 

J (x- a') (x - b') Va 4 



+ bx -\- cj? 

_q -\- a'p P dx 

~ a'-b' J (x-a')Va 



229 



■ f- 

-' (a' 



(x — a') Vo. -\- bx + cx^ 

q + b'p f* dx 

a'-b' J (x - b') -^a-\-bx + cx^ 

dx 



(a' + c'x") VoT 



cx^ 



^ f an-1 ^ ^ («C' - «'«) 



tan""^ X 



-\/a'{ac' — a'c) ^ a' (a + cx"^ 

1 , Va '( a 4- ea-- ) + .r Va 'c — nr' 
log ^ 



230 



2 Va'(a'c — ac') ^a'{a + ^a'"^)- x Va'c — ac' 
(a' + c'a;^) Va + cx^ 



1 .,..-, |o'fa + ex') 



Vc' (a'c — ac') ^ a'c — ac' 

1 1 Vc' (a 4- ex*) — Vac' — a'c 

2Vc'(ac'-a'c) " Vc'(a + cx^) + Vac' - a"c 



231. \fU, \/^n^[^^^ 



where z"(a' + 6'x) = a -\- bx. 



z"~'^dz 



34 MISCELLANEOUS ALGEBRAIC EXPRESSIONS. 



232. ffi^, V c + V'a + bx) dx 



where «" = c + Va + bx. 



where y* (a' + b'x) = a -\- bx and s is the least common multi- 
ple of n, q, etc. 

234. ff(x, ■Va + bx + a;2) dx 

r f2Va-z-b s^Va-bz+Va \ (z^Va - bz +Va)dz 

where xz + Va = Va -\- bx + x\ 

235. ("/(a;, Va + ^»cc + x"^) dx 

- C A '^^~^ V? - bu -V aVl{bu - a - ti^) du 
~J'^\b-2u' 2u-b J {b-2uf ' 



where u = Va + bx + x^ 



x. 



'x2+axV2 + aA „^ _,/aa;V2' 



-'a;^ + a* 4a8V2l W-ax V2+W 
r_rf^ = J_ f lo, f ^::^«A _ 2 tan- Y^^ I 



a^— x^. 



TRANSCENDENTAL FUNCTIONS. 35 



V. TRANSCENDENTAL FUNCTIONS. 



236. j sin x -/(cos x)dx = — \ /(cos x) d cos x. 

237. j cos X -/(sin x) (/a; = | /(sin a-) rZ sin a;. 

238. rsinx-/(siua;, cos x)dx-^- \f{^^ - «^ «)<'^> 
where z — cos x. 

239 r__^^-_ = _l— / f-^-f-^l, 
*^''- J a^h cos a; e(^ - (') \J z ■\- c J z-c j 

where s = tan ^x, and c- = (^^ + «)/(^ - a). [See 651.] 

, , . = 1 , ^; "^, 5' where s; = tan i a;. 

(X ± (^ sm a; »/ a ± 2 fts; + az' 

241. ^/(sin a-) 6?x = -ffUos (^| - ^) ) '^ (^| " ^J' 

242. r/(tan a-) rfa; = - J /ctn f | - xj d (^| " ^ j' 

243. j"/(sec a;) dx = - J/csc f | - a- J fZ (^| ^ ^ j' 

rsmx^f(sui^x)dx^ r J^z) dz 

J Vl-A;^sin2x ^ 2 V(r-^^) (1 - k^^z) 
where z = sin'^a;. 

r cos.r./(cos^x).7.r ^ r.A l-^)^^ . ^here z = sin^x. 
J Vl-A-^sin^a; -^ 2 V^ (1 - A;^ «) 



36 TRANSCENDENTAL FUNCTIONS. 

/' tan X -/(tan^ x) dx _ C J z \ dz^ 

where 2: = sin^ aj. 

247. r/(«x + ^)<^^ = ^ ^ f{ax + S)c?(aa^ + ^'). 

248. rsec" + 2a; ./(tan cr) (Za; = (^(l + z^yf{z) dz; « = tan 3 

249. I /(sin x, cos x) dx 

= — I /( cos I — — X], ?,\n\ — — x]]d\ — — x 



|_^Uin(|-.))c/f| 



//• <i (x) dx 
fix) ■ sin-i x-dx = sin-^ x ■ ^ (a;) — I ^^ ^ _^ > t/ic, 

where <f>(x) = i f(x)dx. 

/C <b (x^ dx 
f(x)-cos-^xdx = cos-^x-<f,(x)+j y ' ^ - 

252. \f{x) ■ t2,n-^xdx = tan-^ x-<f,(x) - j t^ f ' 

253. r/(a;) • ctn-^ ic dx = ctn-^ a; • <^ (a;) + f^M^ . 

254. j /(a;, cos x)dx = — I ./"( o ~ «> sin z j c?2:, 
where 2; = — — x. 

Li 

/' sin a; -/(cos x)dx _ IT J z — a\ dz 
J a-\-bcosx bJ \ b J z 

where z = a + b cos x. 



TRANSCENDENTAL FUNCTIONS. 87 

256. ff(x, ^ogx)dx^jf(6', z)e'dz, where « = log x. 
257_ r fQogx)dx ^ Cf{£^dz, where z = log x. 

258. Cx"'f(log x) dx = Je('" + i>y(s) dz. 

259. (/(sinar, cos a;, tan x, etna;, sec a:, csca;)c?« 

' -J-^Vr+T^' 1 + ^^' 1-^=^' Iz'l-z^' 2z ) 



2dz , ^ ^ 

■ ) where z = tan - ; 



1 + z' 2 



VT^^2 1 1' 






:» where z = sinaj; 



Vi - «' 

= I / , > z, -y VI + s^ 

: J where z = tan a; : 

l-\-z^ 



://(vi, vi^, Vr^^, W' vfe' 7: 



:> where « = sm-'a;: 



2^z(l-z) 



J''\^i + z vi + v^ ^ ^ y 






2V2(1 +«) 



:j where z = tan^x. 



38 TRANSCENDENTAL FUNCTIONS. 

260. I s\nxdx= — cos x. [See 247.] 

261. I sin^ a; c?x = — ^ cos a; sin a; + 1^ cr = ^ a:; — :|^ sin 2 a;. 

262. I sin^ xdx = — -^ cos x (sin^ x + 2). 

«o« r • „ 7 sin"-^a; cosa; . n — 1 f . „ „ . 

263. I sm''xdx = 1 I sin«-^a;c?a;. 

J n n J 

264. I cos a3 c?x = sin X. [See 247.] 

265. I cos^ xdx = ^ sin a; cos x -'c \x = ^x + \ sin 2 aj. 

266. I cos^ a; c?a; = -J sin a: (cos^ x + 2). 

267. I cos" a; c?a; = - cos" ~^ a; sin a; H | cos"~^a;c?a;. 

J n n J 

268. I sin x cos xdx = \ sin^aj. 

269. I sin^ x cos'' a; c?a; = — ^ (:|^ sin 4 a; — x). 

270. I sin X cos"» a; c?a; = — 



cos'^ + ^as 



i.p 



271. I sin"'a; cos a;c?a; = 



in + 1 
sin"» + ^a; 



m+ 1 
272. j cos"" a; sin"x(Za; = 



'•/ 



273. I cos"'x sin"a;rfx = 



cos'^^^a; sin^ + ^a; 

rth -\- n 
m — 1 
m + w. 

sin"~^a; cos'^+^a: 



4- — — r— I cos'""'' a; sin" a; (/a;. 



m + w. 



+ "^ . ^ I cos'"a; sin"-''a:rfx. 



TRANSCENDENTAL FUNCTIONS. 39 

, ri^^m^xdx 1 / sin"-\'r , ^ ^ rsiii"-^a-rf.»'\ 

274. I ■ = \ — \-(n — 1) I 

J cos™ a; n — m \ cos'"~*a; ^ J cos"* a; / 

1 /siu^ + ^-c , , _, rsin"a7(fe\ 

= 7 \ hi — m + Z) I r— 

m — 1 \cos'"-^a; ^ ^J cos"-'*x/ 

_ 1 / sin"-^a; rsin«-2xrfx\ 

~ m — 1 \cos"'^^a; ^'^ ^J gos"'-^xJ' 



rcos,^xdx _ cos^' + ^a- m — ?i + 2 /'cos'"a:;c?cc 

"J siii"a: (/i — 1) sin""-'^ n — \ J sin"~^a; 



'^x m— 1 /^cos'""'^.' 

)in"^^a; m — ?^c/ sin": 

J_ r cos'"-^xd. 
n — lsm"~^x n — lJ sin"~^a; 



(m — n) sin" ^x m — nJ sin" a; 

1 cos'"~^a; m — 1 /*cos"'~^xdx 



. . , _cos™ [tz— x]d[— — x 

, sm^xdx r V2 



/sin'«a:;c?a;_ r 
cos"ic »/ 



sin" ( 77 — a^ 



277. I -^ = log tan x. 

J sin x cos x 

278. ^i- = log tan - + - ) 

J cos a; sin^a; \4 2/ 



— CSC X. 



279. j 



sin^'x cos" a; 
1 1 ,m + w — 2/* dx 



m-\-n — 2 r _ 

3"~^x n — 1 J sii 



n — 1 sin"*" ^ a; -cos"" ^x n — 1 J sin"'a; -cos^^^a; 

1 1 m + n — 2 r dx 

~ m — 1 sin"*~'a; •cos"~^x m — 1 J sin"*"~^a; -cos".-?; 

^ r dx 1 cos a; , m — 2 /* dx 

280. I -. = r -: \ T I • ^ , • 



40 TRANSCENDENTAL FUNCTIONS. 

dx 1 sin X . n — 2 C dx 



C dx 1 sm a- ii — 2 r dx 

J cos"a; n — 1 cos'^^-'a; n — \J cos"~^ 

282. I tan xdx = — log cos x. [See 247.] 

283. I iaxi^xdx = tan x — x. 



284. I tan"a;fZic = r^ — j tan^-^xc^ic. 



285. I ctn xdx — log sin x. [See 247.] 

286. I ctn^xdx = — etna; — x. 

287. jctn^aic^a; = - ^ "_ J^ — jctn"-^xdx. 

o.^^ /^ T -1 i /''■ , »'\ ,1 1+sina; 

288. I sec xdx = log tan I - + - 1 = |- log -t— 

289. I sec^a;c?a; = tan x. 



sin a: 



/r dx sma- n — 2C «^ 
sec^a^rfx = I = ■; TT \ — I I -^^ 
c/ cos"a; (72, — 1) cos" ~^ a; w — l^cos"~^a; 

sin a; , n — 2 C , , 

= 7 ;^ r~l 1 7 I sec"~^a;ria;. 

(n — 1) cos"~^a: n — lJ 



291. I esc a; cifx = log tan -^ X. 

292. I csc'^xc^a; = — ctn x. 



TRANSCENDENTAL FUNCTIONS. 4-1 

293. I csc"a;rfx = | -r-^ 



cos X , n — 2 r dx 



n — 2r dx 

"-^a; n — 1 J sin"-^i 



(n — l)sm"~^x n — lJ sin"~"^ic 

cos x n — 2 f „ , 

= — 7 ..s . „ 1 1 7 I csc^-^xdx. 

(n — 1) sm"~^a; n — lJ 

294. ftr-r^^ = - tan (i tt - i x). [See 241.1 

J 1+ sill X V4 -J / L J 

/dx 
: = ctn a TT - ^ x) = tan (i tt + ^ a;). 
1 — sin a; ^ 

296. 1 :; = tan A x, or esc x — ctn cc. 

»/ 1 + cos X 

/dx 
= — ctn 4 a-, or — ctn x — esc x, 
1 — cos X 

298. f— -^^ = ^-^^ • tan- > (sec ^ • tan i a- ± tan 0), 

if a > i, and b = a sin 6. 

^„„ /* c?.x ± sec a , sin h(a ±x') 

299. I r-^ = 7 — log f-) (» 

J a ± sm a^ o cos -^(x ^ a) 

if 6 > a, and a = 6 sin a. [See 241.] 

««« /* dx — 1 . , r^ + a cos a;~l 

300. I z = , • sm-' — — 

J a -\- b cos X Va^ — b^ \a-\-b cos a; J 



5 



CO ? 

" 5 



o s 
-Sim 



QJ o o 

•^ 5- 



1 . , r Va'^ — b^ ■ sin a;~| 

, sm- — : 

I Q^ — ip- \_ a -\- b cos X J 



b g or : sm- ' I r^ I ■> 



or , 



_ii 1 - ,rVft2-^ 

£s-§ or 



c o 

« •a 



Va* - y 



tan~M y^ tan \ x V> 

, r Vft^— ^ • sin a;~| 

tan-M — T— h 

L c» + a cos X J 



42 TRANSCENDENTAL FUNCTIONS. 



1 , r ^* + <^ COS a; + V6^ — d^ ■ sin x ~\ 
' >2 - a2 °^ L a + 6 cos a; J ' 



V^ 



or 



1 , r V^ + ft + V6 — ft • tan ^ a; ~| 

'-ft' LVZ* + ft-V6-a-tanlxJ' 



1 , , , r V^»2 - ft2 . 
or 



, tanh-T \^^-"^-^^^^ l 

V^2 _ ^2 ^ 6 _|_ (J cos aj J 

_-, r dx 1 ^, , 

301. I — TTT = o , ,i, [b log (ft cos cc + ^> sm x) + ftxl 

J a + 6 tan x a^ + ¥^ ^ /'J 

/f/a; 1 

^ \ = -7= log tan (i cc + i tt). 



r _^mxdx_ _ _ /* COS {\ TT — X) d (^ TT — x) 

' J a -\- b cos x J a + b sin (^ tt — a;) 

= — - log (a + b cos a;). 



304 



/(ft' + J' cos x) dx _ b'x a'b — ah' r 
a + b cos X b b J i 



305 



/ 



+ b cos X b b J a-\- b cos x 

(ft' + b' cos a;) c?a; _ ab' — a'b sin x 
(a + b cos x)^ a^ — h^ a + b cos a; 

ftft' — bb' C dx 



, ftft' - bb' C dx _„ „,.^ 

"• 2 TT I — TT [See 241.] 

ft'' — 6^ c/ ft + ^ cos a; ■- -" 

gQg r (ft' + ^''cosa;)c?x ^ 1 V jab' -a'b)^ix\x 

■J (ft + ^ cos a;)" (w- 1) (ft2- Z*^) |_(a + />cosa-)"-i 

r r(r^^<' - ^>Z'') (» - 1) + (n - 2) (ft/>' - a'b) cos a-] r/x l 
J (ft + 6cosa;)"-i J" 



307./ 



TRANSCENDENTAL FUNCTIONS. 43 

(a' + b' COS x) dx _ (a' — h') tan ^ x 

(1 + cos x)» ~ (2 w - 1) (1 + cos x)"-^ 



n (a' + h') - g.' r dx 

2n-l J (1+ cos x)"-i ' 



308 C——^^— = - r - ^ sin a; 

J (a + b cos xf (n - 1) (a^ - ^^^^ [_(a + Z» cos a;)"-i 

+ (2n-3)aj ^^_^^ 'cos x)— ~ ^'' ~^U (a + b cos a;)'-^ J 

309 f ^^ ^ tanja; 

* J (1 + cos x)" (2 ?i - 1) (1 + cos x)"-^ 

^27i-lJ (l + cosx)»-^ L'e^-'-i^-J 

otn r (*' + ^' COS x) f/a; a'b — ab' , , j . 

310. I ^ — — — '- — - = — ^ -^ log (a-\-b cos a^) 

»/ sin x(a + b cos x) a^ — ¥ ^ ^ 

H — ^ log sin ^x log cos -A- x. 

a + b ^ ^ a — b ^ ^ 

»■... r ((t^' + b' cos x) dx a' ^ , ,, , . 

311. I -^^ — —7-^ — - = -logtan|-a7r + a;) 

J cos x(a + b cos x) a ^ ^ 

(ab'-a'b) f dx 



- a'b) r 
a J 



a -\- b cos X 



«,« r (a' -\- b' cos x) dx 4- («' =F ^') , i / i 7im 

312. \ ,., ^ = ± ,\ ^ +i(a'db&')logtan|; 

J sm a; ^1 ± cos a;) 1 ± cos a; '' ^ '^ * ^ 



313 



/dx _ — ctn I" X 

(1 - cos xy ~ (2 71 - 1) (1 - COS a;) 

n — 1 C dx 



n — l 



4.iL:Ll.r_ ^h-—. ^See241.] 



44 TRAMSCENDENTAL FUNCTIONS. 

dx C dx 



314 



r dx r 
ci^ — b"^ cos?x J (a^ — U^ 



1 . sin (a — x) 

log ^^ : 

2 ab sin a sin (a -\- x)' 



~l "■ — 3"t^^~M~^^ )' whei 
a^ sm /? \sm pj 



1 a 

or -s -: — — tan ' i -: — - j , wnere cos a 



cos /3 b 



„,^ /* tZcc 1 ,/&tanic 
315. I p^ 5 . ,„ . „ =— tan-^i 



a^ cos^ a; + ^^ sin^ a; ab 



a 



316. r44^ = ^^ tan- ftan . . J-^") 
J « + b co&^x b^a \ ^ a + bj 



X 

~b 



<«<» C sm a; cos a:" f/a* 1 , , , , • o s 

J a cos- a; + 6 sura; 2(b — a\ ^ ' 



318. f ^ = C d{x-a) 

J {a -\-b cos X + c sin a.')" J [a + ?• cos (a- — a)]" 

where 6 = r cos a and c = r sin a. 



319. I : [See page 61.] 

J C ' 



dx 

a -\- b cos X -\- c sin x 



-1 



sin" 



J r i^ + c^ 4- a (b cos a; + c sin x) 
^a^ — W—c^ |_ V(&^ + G^){a + ^ cos a; + c sin a;) 

, ^= • log 

V&2 + e^ _ ^2 



] 



[^^ + 0^4- a (& cos a- + csin a;) + V&^ + c^— ^^(isinx— ccosa;)"! 
V(6^ + c^) (a + i cos a: + c sin a;) J 

1 , V^-2 4- ,.2 - a2 _ c _j_ /^ _ (j) tan ^a; 



OP 

o 

' X 



■sIV + c- -a" ■\/b'' + c'-d' + c-(b- a) tan \ 

c^ [_ Va^ -b^-c^ J" 



Va^ - 62 



TRANSCENDENTAL FUNCTIONS. 45 

/dx 1 

— -— — : = - log (a -h c tan |a;) 
a (1 + cos x)-\- c %\nx 'C 

321 f ^^ 

J (a [1 + COS a;] + c sin x)^ 

1 r c (a sin a; — c cos x) i / , . i x ~l 

= -o —^ r-;' ■ a log (a + c tan ix) • 

^r.r. r (a; + sin a?) c^cc ^ , 

322. I ^— r^ — = a;tan-^a:. 

J 1 + cos x 

= |- sin a; Vl — P sin^a; + — sin~^(A; sin a;). 

324. I sin x Vl — A;^ sin^ a; t?a; 

= — -|- cos a; Vl — A;^ sin- x ^-j— log (k cos a; + Vl — A;^ sin^a;), 

325. J sin a: (1 - A;^ sin^ x)^ dx = - i cos a; (1 - A;^ sin^ x)^ 

+ f (1 - k^) jsin X Vl - k"" siu^x dx. 

««„ /^ cos a;c?a; 1 . , ,, . . 

326. I , . ^= = T sin-i (A; sin x), 
^ Vl — k^ sin^x ^ 

or - log (6 sin x + Vl + 6^sin^a;), where b^ = — k^ 

327. I- = — -log(A:cosx + Vl — A-^sin^a;), 
•^ Vl — k^ sin'^ a; ^ 

1 . , Jcosx „ 2 

or — TSin~' , , > where 6. = — k^ 
b Vl+62 

^„„ /• tan a; c?a; 

328. - = 

^ Vl — A;^sin^a; 

1 , / Vl - A;=^ sin^a; + Vl - k''\ 
;log 



2Vi^^ \ Vl - A;2 sin^a; - Vr^A:^ 



46 TRANSCENDENTAL FUNCTIONS 

xdx 



/v cix 
-;——. = - a; tan |a TT - a;) + 2 log cos i (i tt - x). 
1 + sin X ^ 

/cc doc 
' : = X ctn -^ (^ TT — x) + 2 log sin i (i tt — x). 
J. olU X 



331. iz = a; tan 4- cc + 2 log cos -J- a;. 

J 1 + cos X 

332. I :: = — x ctn ix + 2 log sin 4- x. 

J 1 — cos a; 



««« r tan a; fZa; 1 .( ^h — a \ 

333. , ==— =^=cos-M p— -cosa;)- 

^ Vet + ^ tan^ X -\/b — a \ V6 / 

334. r-^^^^- = ^-yU-\^--tan-»( J-.tana- ) . 
J a -\-b tan^ a; a — b\_ ^a \* /J 



««,. r tana^c^x 
335. I 



a + h tan x 

= — — \bx — a log {a + b tan x) + a log sec a; V 

336. I a; sin a;fZa; = sin x — x cos x. 

337. I a;^ sin xdx = 2x sin a; — (a;^ — 2) cos x. 

338. I a;^ sin a; c?a; = (3 a;^ — 6) sin x — (x^ — 6x) cos x. 

339. J x"" sin xdx = — x™ cos x + m % x™~^cos xc?x. 

340. I X cos xdx = cos x + x sin x. 

341. I x^ cos X (fx = 2 03 cos x + (x^ — 2) sin x. 

342. I x' cos X cZx = (3 x'' — 6) cos x + (x^ — 6 x) sin «. 



TRANSCENDENTAL FUNCTIONS. 47 

343. I iC" COS xdx = x'" sin x — in i x'"~^sin xdx. 

.... rsincc , 1 sin ic , 1 Tcos cc , 

344. I dx = ■ —-—J H 7 I ——-7 dx. 

J x"* m — 1 x'"-^ vi — lJx'"-^ 

«.., rcosx , 1 cos a; 1 rsincc , 

345. I dx = T ■ T T I 7«a;. 

J X'" m - 1 a;'«-^ m - 1 J ic'"-* 



346. J c?a; = a; - 77-77^ + ^^^ - ^^^7-, + 



' x^ 

X 3-3! '5-5! 7-7! 9-9! 



,6 a.8 



«>.« rcos X , , x^ , X* a;" , 

3^^- J -^^^' = ^°^^-2:2!-^4T4!-6T6l + 8.8! 

rxdx _ . x^ 7x^ 31a;^ 127 a-« 

^^^- J sinx"'^'^3.3!"^3-5-5!"^3-7-7!"^3-5-9!"^ 



r^^ _ ^ g;^ 5.T^ 61^« 1385 a:^" 

J cosa;~ 2 "*"4-2!"^6-4!'^8-6!"^ 10-8! ' 

/x dx 
. „ = — a; ctn a; + log sin x. 
sin'' x 

351. I — 5— = a; tan X + log cos a;. 

J COS'^X 

362. n^ I aj^'sin^xtfo; 

= a;'"~^ sin"""^ x (m sin x — nx cos a;) 
' + w(?i — 1) rx^sin^-^XfZx — 7;i(w — 1) ra;"'-2sin"a;c?x. 

353, 71^ J x"' CDS'* xc^a: 

_ a;TO-i cos"~^ X (m cos x + nx sin x) 

4- w(w - 1) fx'" cos'^-'^xcZx — m(?» - 1) Cx'"-'^ cos"xdx. 



48 TRANSCENDENTAL FUNCTIONS. 

354. f^ 

1 r a:!"*"^ (m sin x -{- (n — 2)x cos x) 

^ (?i - 1) (?i - 2) |_ sin«-ia; 



365. f^ 

»/ COS" a; 



1 r x^~^(m cos g; — (?i — 2)a; sin a:) 

^ (?i - 1) (w - 2) L cos»-ix 

^ 'J cos"~^x ^ ^J COS" -^ a; J 



^^„ /'sin" re c?a; 
356. J 



X'" 

X ((m — 2) sin x -\- nx cos a;) 



™m— 1 



1 r sin"-^ 

~ (m - 1) (7n - 2) L 

- n'j —;;;=r- + "(« - 1) J ^..-^ J 



«^-. . 'cos" a:c?a; 
357 



1 P cos"~^ a; (na; cos a; — (m — 2) cos x) 

^ (m - 1) (??i - 2) L ^'""^ 

„ /*cos"xfZx , . ., ^cos"~^xdx~\ 

— ^ I 5 \- n(n — 1) I 7, — • 

J a;'"-^^ ^ V a;'"-'' J 

358. I x'' sin"* a: cos"a:c?x 

= a:^"' sin™ a: cos"~^a;(» cosa; -j-(')/i + n)x sin a;) 

(in + 7i)'' |_ \i \ 

+ (n — 1) (//I + ?i) I X'' sin^x cos"~^a:(^x 



TRANSCENDENTAL FUNCTIONS. 49 

— m'p I a;''"^ sin^^^a; cos"~^a;c?ic 

— p(^ — 1) I x^''"^ ^v^'x Q.o^'^xdx • 

= •; — ; — -5 a:^""^sin"'~^a;cos"a:;(» sin a; — (w + w^a; cosa?) 
(m + ny |_ v^ \ / / 

+ (m — 1) (m + 11) I aj^ sin"*"- a; cos" a;c?a; 
-{-np i xP~^ sin"'~^x cos'^'^xdx 

„-n C ■ ■ 7 sin Cm — n) a; sin(m + w)a; ^ 

359. I sin mx sin nx ax = — ^r-^ i ^77 — ; — r~ * 

J Z(m — n) Z{7n -\- 71) 

__- /* . - COS (m — n)x cos(m + n)x g, 

360. I sin mx cos nxdx — tt, r —rr) ; — . ' " 

J 2(m — n) 2 (771 + 71) '^ 

» 

03 

-_- /* , sin (m — w) a; , sin(m + w)a; 

361. I cos waj cos wa; rfa; = — T-7 r 1 777 ; — f— • 

,/ 2 (m — 7i) 2(m + 71) •-' 

362. I sin^ mxdx = tt — ("ma; - sin mx cos ma;). 
»/ 27n^ ' 

363. I cos"^ mxdx = pr — ("mx + sin mx cos mx). 
./ 2 ??i ^ '^ 



1 



364. I sin mx cos mxdx = — -f^ cos 2 mx. 
J 4m 

365. I sin nx sin"'x<Zx = — ; — — cos nx sin^'x 
J m + 71 1_ 

+ m I cos(/4 — l)x -sin^'^xc^x 



50 TRANSCENDENTAL FUNCTIONS. 

366. I sin nx Gos"'xdx = ; — — cos nx cos™ a; 

J m + n [_ 

+ ml sm(7i — l)x-cos'''~^xdx \. 

367. I cosnxsiu^xdx = ■ — sinna; sin'"a; 

J m + n |_ 

-«/sin(„-l)x.sin"-..<^.]. 

368. I cos nx co^'^xdx = ; — sin nx cesser 

J m + n\_ 

+ m I cos (w — 1) a; • cos"'~'icc?a; • 



369 



/cos nxdx _ /^cos {n — V)x dx /^ cos(w — 2)xdx 
cos"'a; "" J cos"'~^a; J cos"'a7 



rcosnxdx __ ^ r sin(?i — l)a;cZa; /* cos (?^ — 2)a-c?g; 
' J sink's; ~ J sin'"~ia; J sin'" a; 

/'sin wa;fZa; _ ^ /^ cos(?i — l)a;c?a; r sin(??- — 2)a;6?a; 
' J sin"'a; J sin'"~^a; J sin"'a; 

/'sin ??,a'fZa; _ ^ /' sin (?i — l)xdx /' sin (n — 2)a;c?a; 
" J cos'" a; »/ cos'"" 'a; J cos'" a; 



^P + n-lfl^ 



/'(cos ;:»a; + i s,\nj)x)dx _ . /';s'' 

■ J cos wa; »/ 1 + ;5;^" ' 

where z = cos a; + t sin x. This yields two real integrals. 



- . /'('cos wa; + i sin «a-)f7a; ^ rzP'^"~'^dz 

374. I ^ ^— ^ = ~ 2 I — — -, 

^ sm na; J 1 — z^"' 

where z = cos a; + i sin x. This yields two real integrals. 



TRANSCENDENTAL FUNCTIONS. 51 

r(icosx — smx)dx_ r dy 
375. J -J 2—;!' 

where y = z^zzm This yields two real integrals. 

V cos nx 



X 



«-,, r ■ • , • 7 , rcos(a — 6 + c) 

376. I sm ax sm ox sin cxax= — x i ^ ; — ; 

J \^ a — b + c 

cos (b + c — a)x cos (a + b — c)x cos (a -\-b + c)x \ 
6 + c — a a + 6 — c a + 6 + c J 

'inn C z. J , fsin (a + ^> + c) 

377. I cos ax cos te cos ca; aa; = i i ^^ — —-. — ; 

J y_ a-\-b -\- c 

sin (6 + c — ft) a? sin (rt — 6 + c) a? , sin (a + b — c)x \ 
b ■\- c — a a — b -\- c a + b — c j 

cos (a -^ b -h c)x 



X 



378. I sin aa? cos bx cos ca; c?a; = — :f i 



cos (6 -f c — g) a! cos (a -\- b — c)x cos (a + c — i) x 



} 



b + c — a a + b — c a + c — b 

««« r -7 • 7 , r sin (a, -F ^» — c) a; 

379. I cos ax sm bx sm ca;ax = f s ^^ — --. — 

J I a -\- b — c 



sin (a — b -h c)x sin (a + b -\- c)x sin (Z* -f- c — a.) a; 
a — 6 + c a -^ b -\- c b + c — a 



380. I sin~^a;c?a; = x sin~"^a; + Vl — x"^. 

381. J COS'^XC/X = X QQS~^X — Vl — x^. 

382. I tan-^xc/'a; = x tan~^x — \ log(l + x^). 

383. j ctn- ^xdx = x ctrr ^ x -}- -^ log (1 + x^). 



x\ 



62 TRANSCENDENTAL FUNCTIONS. 

384. I s.eor'^xdx = x sec~^x — log(x + Vcc^ — 1). 

385. I csc~^ a;c?x = x csc~'a; + log(a; + ■\f^— 1). 

386. I versin-^ icci^a; = {x — 1) versin"' x + V2 a; — 

387. C {?.m-^xfdx = a; (sin- ^ a;)^ - 2a; + 2 Vl - a;^ sin- 'a;. 

388. I (cos-^ a;)^c?a: = x (cos"^ a;)^ — 2 x — 2 Vl — x^ cos~^ x. 

389. fa; sin-^x^x - i[(2x2 - l)sin-ix + x Vl - x"]. 

390. I X cos~^xc?x = i[(2x^ — l)cos~'x — xVl — x^]. 

391. I X tan""'xc?x = ^[(x^ + l)tan~'x — x]. 

392. I X ctn-'xc?x = ^[(x" + l)ctn-'x + x]. 

393. I X sec~'xc?x = ^^[x^ sec""^x — Vx^ — 1]. 

394. I X csc-^xc?x = ^[x^ csc~'x +a^'x'^ — 1]. 

395. I x"sin-'xc?x = — — r ( x" + ^ sin-^x - \ , ^^ \ 
J n-\-l\ -^ Vl - xV 



396. I x"cos-'xcZx = — -— I x" + icos-'x 4- \ ^ , 

J n + \\ J Vl -x^ 



i 



■^>-ck^ ^ ^---e'"' 



^^'■rc^ 



TRANSCENDENTAL FUNCTIONS. 53 



397. Jx"tsin-^xdx = ——(x'' + HEin-^x-Cj 

398. Cx»ctn-'xdx = —^(x" + 'ctn-'x-^ f^!^l^Y 



X / iC 



400 



tan~^a;c?a; , .,.-..„. tan~^a; 



^^ = logo; - 1 log(l + x^) - 



X 



401. Ce'''=dx = —- Cf{e"^)dx=J'^^^^^^, y^e'^. 

a;e«^rfa; = — (ace - 1). 

x^e'^'dx = I x'^-'^e"^dx. 

a a J 

P^ax 1 r e"^ , c^'^d^~\ 

404. I — dx — : + a I ' 

J x"" m — 1(_ x""-^ J a:"'-ij 

J log a (log ay (log a)^ 

n(n-l)(n-2)- • '2.1a^ 
~ (loga)" + i 



a 



bx 



-«« Ta^'^a; 1 r «■" a^- 
407. I — — = T --r - 7 



log a 



2)a;»-2 

a^ • (log ay 



+ 



(w-2)(ri-3)a;"-3 (w 



(logq)"-^ r a='dx ~\ 

- 2) (n- 3) "-2.1 J x J" 



,^„ Ca^dx , , , , (cc log a)^ , (a; log a) 



3 

+ 






J 

54 TRANSCENDENTAL FUNCTIONS. 



409. I :: = Ior i ^-^^ ' ■*- 

J 1 + e^ '' 1 + e^ 

/dx 1 
411. \ —z;z-^-, — -: = ^=tan-M e^-^A/Tr 



412. )— ^^== j=\\og(-\/a + be""'-Va) 



/ /— 2 -\/n 4- /)^'"^ 

- log (Va +6 e"- + V^) L or == tan- ' ,Z_1 ^ 

VI -y/ZTa V- a 

J {\+xf 1 +x J a(n + 1) 

A-iA C ^^ 7 e'^ia sin ?;.x — » cos px) 

414. I e"" sm »a? (/a? = — ^^ -^ f ^-^• 

Atn C r,^ 7 ^'^ (f' cos px + » sin »a^) 

415. I e"" cos »x- dx = — ^^ ^, — ^ ^-^ • 



416 



e*" log a; c^a; = 2 I 

a aJ X 



goa: gij^2 xdx = J— — ^ ( sln X (o, B\Xi X — 2 cos cc) + - ] ' 

/e'" / 2\ 

goa; cos'^ajc^a; = — — — - ( cos a; (2 sin x + a cos a;) + - 1 • 

419. I e'^sin"6xc?x = -5— — ;-^( (a sin bx 
J a^ + tr¥ \ ^ 

— nb cos bx) e'^ sin"~^ Jx + n (n — l)b- i e^ sin"~^ bxdx 



TRANSCENDENTAL FUNCTIONS. 



55 . 



). I e'" cos" bxdx = -^—, — ^-^ ( (a cos bx 
J a^ -\- n-h' \ ^ 

+ nh siu te) e"^ cos"~^ hx -\- n {n — X)h^ I e°^ cos"~26icc?a; )• 



421. re^^tan^ajc^ic 



n 






/' 



e'^tan"-ixc?a; — | e'^tan"-2a;c?x 



422. re''^ctn«a;(Za; 
e'^ctn"-!^ 



423 



/ 



n-1 



e"^ dx 



+ 



a 



n 



-J 



-/' 



e'^ctn''-^xdx— | e'" ctn"-^ a; c?a;. 



a sin X -\-(n — 2) cos a? 

pOX V i 

sin" X {n — 1) {n — 2) sin''"^ x 



+ 



a^ + (71 - 2y re^^dx 



(n - 1) (7^ - 2) 



-2) J sin"-^ 



x 



424. 



/ 



e"^dx 



^ a cos X —(n — 2) sin a; 
cos" a; "^ (n — 1) {n — 2) cos""~^aj 

a^ + (71 - 2)^ r e'^'dx 



+ 



-2^ f 

-2) J ( 



(» — 1) (ti — 2)»/ cos""~^a; 

426. I e"^ sin"' a; cos"xdx 

= 1 ; — TT"^ — ^ 1 ^""^ sin"^ X cos"~^ x (a cos x + (m + n) sin a;) 

(w + ny + a^ K. ^ \ / / 

— 7na I e'^sin"'~^a; cos"'~^a;(^a; 

+ (?z. — 1) (m + n) I e"^ sin"' a; cos"~'^a;c^a; V 



66 TRANSCENDENTAL FUNCTIONS. 

= ~, ; — r^~. — :, \ e"^ sin"*"^ x cos" x (a sin x — (m -\- n) cosx') 

(m + ny -\- a^ {. ^ ^ ' ' 

■\- na \ e'^sin'"~^x cos"~^icc?a; 

+ (m — 1) (to + ii) I e"^ sin"*"- x cos" xc^x \ 

= •:^ 7^; -c\ re"^cos"""^a;sin"'~^a;('asina;cosic + ?isin^a; 

(m + ny -\- a V 

— mcos^a;)] + 7t(w — 1) | e"^ sin'"cccos"~^icc?a; 
-\- mim — V) I e"^ sin'"~^ X cos^iccZo; [- 

= -A re"'^sin"'~^£CCOs"~"^a;(asinxcosa:+ wsin^x 

— m cos^x)] + w(w. — 1) I e"^ sin'"~^iccos"~^a;<Za; 

+ (m — n) (m + n — 1) j e"^ sin"'"^ x cos" xc?x V 

= -^\ f(e''*sin"'~^a;cos"~^a;(asinxcosa; + nsin^x 

{m + ny + a^ [^ ^ 

— mcos^a;)] + m(m — 1) j e^^sin^^^xcos^^-xc^x 

— (m — w) (w + w — 1) I e'"sin'"x cos"~^xo?x V . 

426. I log xdx = X log X — X. 

427. rx'"logx(Zx = x"' + ir^^^-- — ^^--^1- 

428. j (log xydx = X (log x)" — 7i j (log x)"-^ c?x. 

/x'""'' Vlos xV 7? /* 
x™ (log x)" fZx = \ V T I a;'"(logx)"-'rfx. 
^ ^ m. 4-1 m + l»/ 



TRANSCENDENTAL FUNCTIONS. 57 

(log xydx _ (loga;)""'"^ 



430. f - ^. 

J X /i + 1 



^3lJl|^-log(log.)^Iog.^(^V(|ifV 

432 f ^^^ = ^ + _^f^ 

J (log xy {n - 1) (log xy-^ n-lJ (log a;)"-' 

r x"'dx a:"' + ' w +1 T a;"'c?a; 

J (iog^~~(»i-l)(logx)»-i ?i-lJ (ioga:)"-i" 

■- — ^ = I dy, where ?/ = — (m + 1) log x. 

log a; J V 



435. r ^^ -\o.(\o<rx) and f C^^-D^--^ ^ -^ 

J xlo^x- ^°^ ^^°^ ''^' J X (log X)" (log a;)»-i 

436. flog («2 + cc2) (^a- = a; ■ log {a^ + x'')-2x + 2a- tan-' (^ ^ 

437. j (a + Ja?)'" log a;6?a; 

438. j a;"' log {a + to) o?a; 

1 r /'a^'" "•" ' dx~\ 

= ^^M^T r"" ^"^^" ^ ^"^ ~ ^ J "^Tto J * 

/^log (g + ^3^) dx 



439 

X 



, to 1 fbx\ , 1 /toV 
= loga.logx + --2i(^-J +3i(^-; - 



68 



TRANSCENDENTAL FUNCTIONS. 



r \ogxdx 
**"• J (a + bxy 



441 



_^ 1 r log a; r dx "I 

6 (m — 1) |_ (a + te)"'-i J cc (a + te)"'-ij 

/loa: xdx 1 , , , , ^ 1 /*log (a + ^ic) dx 

— ^ = ^logx.log(« + fa)-J X 



442. I {a + bx)\ogxdx=- — log a; ^-f ax — :^to^ 

443. I ,^ 

= - (log a; — 2)Va + bx +Va log(Va + 6x -fVa) 
— Va log ( Va + 6a; — Va) , if a > 

= - [(log a; - 2)Va + 6a; + 2V^ tan" ^ ^^^^^^ 1 .if a < 0. 

444. I sin log xdx = ^x [sin log a; — cos log a^]. 

445. I cos log a;c?a; = \x [sin log x + cos log a;]. 

446. I sinh xdx = cosh x. 

447. I cosh xdx = sinh a;. 

448. I tanh a; c?a; = log cosh a;. 

449. I ctnh xdx = log sinh x. 



TRANSCEIJTDENTAL FUNCTIONS. 59 

450, I secli xdx = 2 tan~ ^ e'. 

451. I csch a;c?a; = log tanh -• 

/I . % — 1 /* 
sinh"x(Zx = -sinh"~^a;-cosh a; I siuh"^^ xc?a; 

sinh" + ^ a; cosh cc — r I sinh" + ^ a; ^x. 



7i + 1 w + 



/I . w — 1 /* 
cosli"xc?a; = -sinha:- cosh""' a; H I cosh"~2a;^x 
w n J 

= sinh a; cosh" + ' a; H -^ | cosh" + ^ a; c?a;. 

n + 1 n-^lJ 

454. I x sinh xdx = x cosh a; — sinh x. 

455. I a: cosh xdx = x sinh a? — cosh x. 

456. j a;'' sinh xdx = (x' + 2) cosh a; — 2 a; sinh x. 

457. I a;" sinh xdx = x" cosh a; — wa;""^ sinh a; 

+ n(n — 1) I a;"~^ sinh xdx. 

458. I sinh^ a; c?a; = |^ (sinh x cosh x — x). 

459. I sinh a; • cosh xdx = \ cosh (2 a). 

460. I cosh^ a;(Za; = | (sinh x cosh a; + a;). 

461. I tanh^a:c?x = a; — tanh x. 



60 TRANSCENDENTAL FUNCTIONS. 

462. I ctnh^ xdx = x — ctnh x. 

463. j sech^ xdx = tanh x. 

464. j cscli^ic c?a; = — ctnh x. 

465. I sinh~^ xdx = x sinh~' x — Vl + x^. 

466. I cosh~^ a;c?a; = x cosh~^ x — Va- — 1. 

467. j tanh- ^ cc (Za; = x tanh"^ a; + |- log (1 — a;^). 

468. Cx sinh-i rrt/cc = ^[(2 x^ + 1) sinh"^ x - cc Vl + x^]. 

469. I ic cosh-^ a;6?a; = :J[(2a;^ — l)cosh-^ x — xVx^ — 1]. 



* ./ cosh a 



+ cosh a; 

= csch a [log cosh ^ (a; + a) — log cosh ^ (a; — a)]. 
= 2 csch a • tanh-^ (tanh ^ a- • tanh ^ a). 

/dx 
; -. — = 2 CSC a ■ tan- ^ (tanh 4- a; • tan i^ a). 
cos a + cosh X \ ^ ^ / 

/dx 
- — \ ; — = 2 CSC a • tanh" ^ (tanh ^ x ■ tan A a). 
1 + COS a ■ cosh X \ i ^ / 

473. j sinh x ■ cos x c?a; = ^ (cosh x ■ cos a; + sinh x ■ sin x). 

474. I cosh X • cos xdx = ^ (sinh a; • cos x + cosh a; • sin x). 

475. j sinh x- sin a;c?a; = ^ (cosh a; • sin x — sinh x • cos x). 



TRANSCENDEiSTTAL FUNCTIONS. 61 

476. I cosh X • sin xdx = \ (sinh x • sin x — cosh x ■ cos a). 

477. I sinh (ma;) sinh (wa;) c?a; 

= — ^ 7, m sinh (?ia;) cosh (pix) — n cosh (jix) sinh (mo;) • 

478. I cosh (mx) sinh (wa;) dx 

= — ^ J m sinh (na;) sinh (wa;) — n cosh (wa;) cosh (vix) • 

479. I cosh (mx) cosh (raa;) 6?x 

= - 2 _ — ^ m sinh (ma;) cosh (wa;) — n sinh (/ia;) cosh (mx) ■ 

/ ■ dx _ r (Z(tana-) 

a cos''^ X -\- c sin x • cos x -\- b sin^ a; J a + c tan a; + 6 tan"'^ ar 
/(I + 7/1. cos a; + w sin x) dx _ T (m cos 8 + ?i sin 8) cos s • dz 
a -{- b cos a; + (^ sin x J Z 

' I ■ dz r (7)1 sin 8 — n cos 8) sin s ■ dz 



+ . , . ^ 



where b — q • cos 8, c = q- sin 8, s = a; — 8, Z = a. + y • cos 2;. 

C . , , X • V , 7^ 7 [See 303 and 304.1 

I sm (mx + a) • sm (na: + 0) aa; ^ -^ 

sin [?«a; — nx + a — &] sin [?/ia; + '^^ + <* + ^] 

~ 2 (??i — 7i) 2 (?/i + «) 

I cos (mx + a) • cos (iix + i) rfa; 

sin [^mx -{- nx -\- a -\- b"^ sin [mx — nx -\- a — b"] 

^ 2 (wi + n) ^ 2 (??i - n) 

I sin (mx -\- a) ■ cos (nx + h) dx 

cos [ma- + wa^ + «■ + ^] cos [^mx — nx + a — b~\ 

~ 2 {m + w) 2 (?/i — w) 



62 MISCELLANEOUS DEFINITE INTEGRALS. 



VI. MISCELLANEOUS DEFINITE INTEGRALS* 



481. J x''-'^e-''dx= j log- dx = T(n). 

T(z + l)=z-T(z), if z>0. 

r(jj)-T(l-y)=^yiil>y>0. r(2)=r(l)=l. 

r (w + 1) = n I, if n is an integer. T (z)= Il(z — 1). 

r(i) = Vt^. Z(y) = i)^[log r(t/)]. Z(l) = - 0.577216. 

>,oo C w^ X ,, r" a:'"-^^/:^- T(m)T(n) 

482. a:'"-'(l -a;)"-irfa;= I -— — — — = ^\ , .-^ • 



483. I sin"a;c?a;= ) cos"xdx 

%/o «/o 



484. 



1-3-5 •• -(71-1) IT .^ . . , 

= o A r> — . s -77' if w IS an even integer, 
J • 4 • D • • ' in) Ii 

= ■ ^ _ ^j if 71. is an odd integer, 

= \ Vtt — ^ {-•) for any value of n greater 

rTl + lJ than-l. 

J'^'^sinmxc?ic 7r.„ ^^^.„ ^ 7r.„ ^. 
= -) if m>0; 0, if m = 0: — — > it ??i<0. 
a; .^ J 



* For very complete lists of definite integrals, see Bierens de Haan, Tables d'inti- 
grales (Ufinies, Amsterdam, 1858-64, and Nouv. Tables d'intigrales difinies, Leyden, 
1867. 



MISCELLANEOUS DEFINITE INTEGRALS. 63 

,^^ /^* sin x» COS ma;c?a; ^ .- ^ ^ ^ ^ 

485. I = 0, if w<- 1 or m>l; 

»/0 X 

— > ifw = — 1 or m = l: — > if — l<7n<l. 
4 Z 

.__ C^ &Ui^xdx TT 

486. I ^ — = TT • 

Jo x^ 2 

487. J cos{x^)dx = J siii(a;2)(Zx = i\|-- 

sin A;a; • sin mx dx = \ cos A:a; • cos mx dx = 0, 

*/o 

if A; is different from m. 

489. I sin^ mxdx= I cos^mccc^a; = jr* 

.-- r'^ COS mxdx TT ^ ^^ 

»/o 1 + a;^ 2 

.-, f"^ eosxdx C^ smxdx ir 

491.1 ^z^= I -- = ^-. 

492. r"e-"'^^x = ^V^- = j^ra). 

c/o 2 a 2(z ^^-^ 

493. I a;"e-°^c?a;= ^ ^. ^ =^-7- 

.«>. r" , » , 1-3-5- • •(2?i-l) Pr 

Jo 2" + ia» ^a 

e ^dx = ^ ^^ - a>0. 

496. I e-"^ VxcZa; = 77- \/- • 
«/o 2 n ^ 7i 

497. f"^c^a; = V-- a>0, 
Jo Vx ^ ^i 



64 MISCELLANEOUS DEFINITE INTEGRALS. 

dx IT 



498 









•f 

499 r*__^^^_^ 

sinli (ma;) • sinh (nx) dx = \ cosh (ma;) • cosh (nx) dx 

= 0, if m is different from n. 

cosh^ (mx) dx = — \ sinh^ (mx) dx = 
502. I sinh (mx) dx = 0. 
cosh (mx) dx = 0. 
sinh (mx) cosh (wx) c?a; = 0. 
sinh (mx) cosh (mx) c?a; = 0. 



506. I e~ "^ cos mx dx = -5— ; ? if a >■ 0. 

a^ + m^ 





a-* + m^ 



J'' m 
e-"^ sin mxdx = -r-; ;» if a > 0. 
a-* + m"^ 

6-'^'=^ cos hxdx = -^ a>0. 

Za 



••X'l^ 



509. I ^:^^^c;a; = -^- 
x o 



510. rM£^=_^. 

511, r'J<^,&=-^^ 

»/o 1 — a;'' 8 



MISCELLANEOUS DEFINITE INTEGRALS. 65 

1 + x\ dx ir^ 



512. f\og(l±^).^ = 

»/o \1 — X/ X 

513. r^i^l^ = -?log2. 
Jo Vl-a;2 2 

515. J (loga;)"cZa;=(- l)«-w!. 
516.X'(lo.i)'.. = ^. 

518. f , '^ = V?. 

519. ra:"'log(-)(Za:= T'^''t,2i >^f^ + 1>0, ^ + 1>0. 

520. flog ('?^V = T- 
Jo ^ \e^ - ly 4 

jr IT 

log sin xdx = \ log cos a;c?aj = — — • log 2. 
.0 c/o 2 

X ■ log sin xdx = — — log 2. 
523. I log (a±b cos x)dx = tt log ( ;r ] • « ^ 6. 



66 ELLIPTIC INTEGRALS. 



VII. ELLIPTIC INTEGRALS. 



d6 r^ dz 



where k^ <d, x = sin <fi. 
E {4>, k)=f Vl - k^ sin^ e ■ dO. 

y 

(1 + ri sin^ ^) Vl - k' sin^ ^ 

<^ = am u, sin <^ = cc = sn m, cos <f> = Vl — aj^ = en m, tan (f> = tnu, 
A<t> = Vl - A;2 sin^ <^ b Vl - kV = dn «, A;'^ = 1 - A^l 

t< = am~^(<^, ^)=sn~^(a;, A;)=cn~^(Vl — x% k) 
= dn-i(Vl-;k2^2, A;). 

JS:=i^(i7r, ^), K'=F(i7r, k'), E=E{^'ir, k), E'-E^Tr, k'). 

T* 7 2 A;* sin 2 (0 

It ko = q — — 7 and tan <^ = 



1 + k k + cos 2 <i) 



524. r '' 

»/0 



Vl - k^sin^e 



= I [i + (i)'.' + [^) k' + (ifl)"^ +.,.], it ,. 



<1. 



= K. 



525. Vl- k'sin'^ede 



). J Vl 

=i[-(«--(i^yf-G^yf--]""'<^- 



ELLIPTIC INTEGRALS. 67 

,474 X'O'O A -J c. I I 

= J'\<i>, k), 

3 5 5*3 

where VI4 = i sin^ «^ -f- — , ^ = ^ sin* <^ + — sin^ <^ + ^^^, 

A = isin«<^ + g^sin*<^ + g^sin^<A + 3^g^ 



. r* Vl - k^ sin' 6- dd = - <i>- E -\- Bva <!> cos, A ~k' 

1 



= E{4>, k). 



J , = sn-i(aj, k) 

V(l - a;2) (1 - k'^x') 



528. . , 

V(l - x") (1 - k'^x') 

= F{sh\-^x,k). Q<x<l. 

529. r , '^"^ =cn-^(g, A;) 
»^^ V(l - x^) (A;'2 + Ji-x^) 

= F{cos-^x, k) = sn-i ( Vl - x% k). 0<x<l. 

530. C , "^^ =dn-i(a;, A;) 

= #(A-ia;, k) = sn-^ Q Vl - x% k\ < a; < 1. 

531. r , ^"^ =tn-\x,k) 
-'o V(l + X') (1 + A;'2a;2) 

= i^(tan-'a;, A;) =sn-Y--=^=' A; Y < cc < 1. 

\ Vl 4- cc^ y 

* The next forty-two integrals are copied in order from a class-room list of Prof. 
W. E. Byerly. 



68 ELLIPTIC INTEGRALS. 

532. r . ^^ =2sn-U^,k) 
^0 -Vx (1 -x)(l- Jc'x) ' 

= 2F(sm-Wx,k). 0<a;<l. 

533. r—==J^== = 2 cn-i ( V^, k) 
^^ Vx (1 - x) (k'^ + k^'x) ^ ^ 

= 2 i^(cos-i V^, k) = 2 sn-i(Vl-cc, A;). < a; < 1. 

534. r ^ ^^ = 2 dn-i ( V;, A;) 

•^^ Vx (1 - a;) (x - k'^) ^ [ 

= 2i^(A-i V^, k) = 2 su-i Q Vl -x,k\- 0<a;<l. 

535. r , "^^ =2tn-^(V^, /^) 
^^'o V(l + a;) (1 + A;'2a;) ^ ^ 



= 2i?'(tan-iV^, A;) = 2sn-Y^-^,>tY 0<a:<l. 

536. r , ^^ ^lsn-Yf,-^Y a>6>a;>0. 

537. r , ^^ ^lsn-Y^,^Y ^>a>^. 



538. . 

^x V(a2 4- x") (b^ - x") 



cn-if?, ^=i=Y b>x>0. 



V^M^' V^ V 



539. J^ ^^ 






J -a 



ELLIPTIC INTEGRALS. 69 

dx 



V(a* - a;2) {x' - b^) 



1 ,/ \a^ - x^ la'' - b^\ 



541. r ^^ 



'" V(x2 + ay(^M^') 



542. X' ''^ 



V(a; - a) (x - y8) (a; - y) . 



y 






Va-y K^^-y ^a-y 



V(a; — a) (cc — /3) (a- — y) 
2 



Va^ 



y 



(V!^;- V!5^) 






'^ V(a — x)(x — /3) (a; — y) 



545, 






Va — y \^«-^ 

c?a; 



V(a — a;) (a; — )S) (x — y) 



2 ^..-i/'./^Lziy i^^^. J^-/3^ 



Va-y V^«-/5^-y ^«-y. 

c?a; 



V(a^^^) (/? — a;) (a; — y) 
2 



Vo-" 



a > a; > 6. 



1^ ,/a; /a^ - b'\ 

a > /3 > y. 



^sn-Ur^, V^^^V a.>a. 



sn-M \ ^' \r^ ^ • x>a. 



' --('^/^»■ Vfi|)- <'>->^- 



sn-i(-V ^ ^' -V ^ )• a>a->^. 

Va — y 

546. f^ 



\ iS — y a — a; ^a — yj 



70 ELLIPTIC INTEGRALS. 

'^ dx 



5«X 



y V(a — x){fi — x) (x — y) 
2 



sn" 



548 






Va — y 



■(^§i^. V!5^)- ^>^>y 



^ V(a — cc) (/3 — cc) (y — a;) 



Va 
549. f 



7 



(V^.- >/^) 



sn-^UL:^, X^^-^ • v>a;. 



V(a-x)(^-a^)(y-a;) 
2 , / /a — y 



Va — V \ ^a — a; 






sn-M \ ^' \ ^ • y>x 



550. 



X' 



a > y8 > y > 8. 
dx 



V(x - a) (cc - /3) (a; - y) {x - 8) 
2 



, / 113-8 X -a l/3-y a-8\ 



V(a-y)08-8) 

a;>- a, 

531. f"- "^^ 



V(a. — a;) (a; — /3) (a:; — y) (x — 8) 



\^a — j8x — 8 ^a — 



^ y-8^ 



552 



V(a - y) (/3 - 8) \^a--^x-8 ^a-y/3-8j 

a>x> fS. 



■X 



^ V(a - a;) (a; - ^) (a; - y) (x - 8) 



\^a— ^x— y ^ a 



2 _g^-ir J«zi^y ^-^. J«^i^ 1^ 



V(a - y) ()8 - 8) \^a-^a;-y >'a-y^-8y 

a > a; > /3. 



553 



X 



ELLIPTIC INTEGRALS. 71 

^ dx 



^ -si {a - x) (/3 - X) {x -i){x- 8) 

V^/3-y a-x' ^a-y ^-l) 



2 

sn" 



V(a - y) (/? - 8) 

^ > CC > y. 



554. J''^ "^^ 



^ V(a - CI-) (^ - a;) (cc - y) {x - 8) 



2 



sn" 



V^)8-y a;- 8 ^/a 



;8-y o^^ 



■V(a -y)()8-8) V^^-y^-S ^a-yj8-8^ 

i8 > a; > y. 



555. f^ "^^ 



V(a -x){(3- x) (y - a;) (x - 8) 



.2 s,,-irj^^ y^^. J^^:il y-8 



Va 



V(a -y)(/3-8) V^y-S^-a; >'a-y^-8y 

y > a > 8. 



556. J^ ^^-^ 



's V(a - a;) ((8 - x) (y - x) (a; - 8) 



\ *y — 6 a —X ^a 



1 .^-if.l^nj^^^^^. J^l^lI y-g 



V(a-y)(^-8) \^y-8a-x ^a-y(i-8j 

y>x>8. 



X^ (Ir 



V(a - x){fi- x) (y - a;) (8 - a;) 
2 



sn" 



i/^J^-y.g-^ /)8-y a-8\ 
V^a-8 y-a;' ^/a - y )8 - 8/ 



V(a-y)(/3-8) 

8>a;. 



558. i sna;c?x = - cosh~M -yp ]• 

559. I en a; c?a; = - cos~^ (dn a;). 



72 ELLIPTIC INTEGRALS. 

560. I dn xdx = sin~^ (sn x) = am x. 

561. r-^^^iogf" 'Y^ 1- 

J s,nx [_cn X 4- dn a; J 

^nn r (^^ It F^' SH 0? + dll X~\ 

562. I = - log • 

J cnx k' \_ en ic J 

_„„ /* c?aj _1 J FA;' sn X — en a?"] 

' J dn X k' \_k' sn ic + en xj 

sn^xdx = Ti[a^ — -E^(ania;, A;)], 

565. j en^ajc^cc = — [^(ama;, A;) — k'^x"]. 

566. I dn^a;c?a; = ^(am x, k). 

567. (m + 1) fsn'^ajc^x = (m + 2) (1 + A;^) fsn'^+^a^cfaj' 

— (m + S)k^ I sn^ + ^a^c^a; + sn'" + 'a; en a; dna;, 

568. {m + l)k'^Ccn"'xdx = (m + 2) (1 - 2 k^)Ccn"'+^xdx 

+ (m + 3)A;M cn'"+ *a;c?a; — cn'"+^a; snxdnx, 

569. (m + 1)^'2 rdn"»a;c?a; = (m + 2) (2 - k^)fdn'" + ''xdx 

— (m + 3) j dn"*+*a;c?x + A;^dn"*+^a;snaena;, 



Since sin2 ^ = _ _ _ (i _ A;2 • sm2 ^), 



J ^2 sin2 6id& \ r^ dd 1 /^2 ^ 

VI -A;3sin2(? ^Vo Vl _ A;2sin2(? *Vo 



TRIGONOMETKIC FUNCTIONS. 73 



Vm. AUXILIARY FORMULAS. 



A. — Trigonometric Functions. 

570. tan a • ctn a = sin a • esc a = cos a ■ sec a = 1. 
tan a = sin a -j- cos a, sec^ a = 1 + tan^ a, 
csc^a = 1+ ctn^a, sin^ a + cos^ a = 1. 



571. sin a = V 1 — cos^ a = 2 sin ^ a ■ cos ^ a = cos a • tan a 



fe=Vi 



1 tana /I — cos 2a 2tan4-a 



Vl + ctn^a Vl+tan^a ^ 2 l + tan^^a 



=v 



gpp- ^ "1 

= ctn ^a • (1 — cos a) = tan -^ a • (1 + cos a). 



sec^a 



572. cos a = Vl — sin^ a = = = = 

Vl + tan^ a Vl + ctn^ a 



-4 



1 + cos 2 a 1 - tan^ ^a „ , . „ , 

~?^ = -. , ^ — rf — = cos^ia — sin^ia 

2 1 + tan^ ^ a ^ ^ 



= 1—2 sin^ |- a = 2 cos^ ^ a — 1 = sin a • ctn a 
sin 2 



a _ /csc^ ^ ~ 1 _ ^^^ i ^ — ^^'^ "2" ^ 
a * csc^ a ctn 4- a + tan 4- a 



-„n ^ sin a Vl — cos^ a sin 2 a 

o7o. tan a = 



Vl — sin^a cos a 1 + cos 2 a 

1 — cos 2 a _ /l — cos 2 a _ 2 tan ^ a 
sin 2 a ' 1 + cos 2 a 1 — tan^ ^ a 

sec a _ 2 _ 2 ctn |^ a 

esc a ctn \ a — tan ^ a ctn^ ^ a — 1 



74 



574. 



TEIGONOMETEIC FUNCTIONS. 



1 — 


— a. 


90° ± a. 


180° ± a. 


270° ± a. 


360° ± a. 


sin 


— sin a 


+ cos a 


T sin a 


— cos a 


± sin a 


cos 


+ COS a 


T sin a 


— cos a 


± sin a 


+ COS a 


tan 


— tana 


T etna 


± tana 


T etna 


± tan a 


ctn 


— ctn a 


T tana 


± ctn a 


T tana 


± etna 


sec 


+ sec a 


T CSC a 


— sec a 


± CSC a 


+ sec a 


CSC 


— CSC a 


+ sec a 


T CSC a 


— sec a 


± CSC a 



575. 





0°. 


30°. 


45°. 


60°. 


90°. 


120°. 


135°. 


150°. 


180°. 


sin 





i 


i^ 


iVs 


1 


iV3 


iV2 


i 





COS 


1 


iVi 


iV2 


i 





-i 


-iV2 


-IV3 


-1 


tan 





1 

V3 


1 


V3 


00 


-V3 


— 1 


1 
V3 





ctn 


CO 


V3 


1 


1 

V3 





1 

V3 


—1 


-V3 


00 


sec 


1 


2 
V3 


V2 


2 


CO 


-2 


-V2 


2 
V3 


-1 


esc 


CO 


2 


^y^ 


2 
V3 


1 


2 
V3 


^ 


2 


00 



576. sin ^ a = V^(l — cos a). 



577. cos ^ a = V^(l + cos a). 

578. tan ^ a = ^— 



cos a 



cos a 



sm a 



+ cos a sin a 1 + cos a 

579. sin 2a = 2 sin a cos a. 

580. sin 3 a = 3 sin a — 4 sin^ a. 

581. sin 4 a = 8 cos^ a • sin a — 4 cos a sin a. 



TRIGONOMETRIC FUNCTIONS. 75 

582. sin 5 a = 5 sin a — 20 sin^ a + 16 sin* a. 

583. sin 6 a = 32 cos* a sin a — 32 cos^ a sin a + 6 cos a sin a. 

584. cos 2a = cos^ a — sin^ a = 1 — 2 sin^ a = 2 cos^ a — 1. 

585. cos 3 a = 4 cos^ a — 3 cos a. 

586. cos 4 a = 8 cos* a — 8 cos^ a + 1. 

587. cos 5 a = 16 cos* a — 20 cos^ a + 5 cos a. 

588. cos 6 a = 32 cos^ a — 48 cos* a + 18 cos^ a — 1. 
2 tan a 



589. tan2a = 

590. ctn2a = 



1 — tan^ a 
ctn2 a - 1 



2 ctn a 

591. sin (a±ft) = sin a • cos /? ± cos a • sin )8. 

592. cos (a± fi) = cos a • cos yS =f sin a • sin ft. 

..«« , ^N tan a ± tan 5 

593. ta.n(a±ft) = - — ^• 

^ '^^ 1 rp tan a ■ tan /3 

..«.. , ^x ctn a • ctn )S rp 1 

594. ctn(a±/5) = — ^ ^ Z, ' 

^ ^^ ctn a ± ctn /3 

595. sin a zt sin )8 = 2 sin ^ (a =b /3) • cos i(a + iS). 

596. cos a + cos /8 = 2 cos ^(a + /8) • cos |(a - /3). 

597. cos a - cos /8 = - 2 sin |(a + /3) • sin \{a- ft). 

sin ("a d= S) 

598. tana±tan/3 = ^ ^• 

cos a • cos ft 

-~^ « sin (a ± ft) 

599. ctn a ± ctn ;3 = ± ^-^^ — r^- 

'^ sin a- sin /3 



76 TRIGONOMETRIC FUNCTIONS. 

_„_ sin a ± sin yS , , , _, 

600. ■ ^ = tan i (a i S). 

cos a + cos p ^ ^ 

sin a dz sin /3 

601. ^ = — ctn i (a + S). 

cos a — cos /8 i \ » / 

„-_ sin g + sin ^ _ tan -|- (a + j8) ^ 
sin a — sin /3 tan ^ (a — j3) 

603. sin2 a - sin^ ^ = sin (a -\- (3) ■ sin (a - (3). 

604. cos' a - cos' ^ = - sin (a + /3) • sin (a - /3)c 

605. cos' a — sin' /3 = cos (« + /3) • cos (a — /3). 

606. sin xi = ^ i(e^ — e~^) — i sinh x. 

607. cos xi = ^{e^ + e~^) = cosh x. 

608. tan xi = -^^ — ; — i tanh x. 

6^ + e~^ 

609. e^+ 2'' = e^ cos ?/ + ie^ sin ?/. 

610. a^ + 2'' = a=^ cos (2/ • log a) + ta'^ sin {y • log a). 

611. (cos zti- sin ^)" = cos nd ±i- sin n^. 

612. sin a; = — i i(e" - e"^). ^ 

613. cos a; = I- (e^" + e""). 

614. tan x = — i -z—. • 

e-^ + 1 

615. sin (x ± yi) = sin x cos yi ± cos x sin yi 

= sin X cosh y ± * cos x sinh y. 

616. cos {x ± yi) = cos X cos 3/1 q= sin x sin yt 

= cos X cosh y 4= » sin x sinh y. 



TRIGONOMETRY. 77 



617. 



In any plane triangle, 
a b c 



sin A sin B sin C 

618. a'^ = h''^-c''-2bc(toQA. 

„-_ a-\-b _ sin ^ + sin B _ tan ^ (A -\- B) _ ctn ^ C 
' a — b sin .4 — sin B tan ^ (.4 — i?) tan ^ (.4 — 5) 



620. sin^^=^^^ — ^^ — ^, where 2s = a + 6 + c. 



621. cosM=^pZ«). 



622. tani^^>~/^^^;^> 
^ >' s (s — a) 



623. Area = ^ ic sin ^ = Vs (s — a) (s — 6) (s — c). 



In any spherical triangle, 

-„. sin A sin B sin C 

624. -^ = —. — - = — 

sm a sin b sm c 

625. cos a = cos 6 cos c + sin 5 sin c cos A. 

626. — cos ^ = cos B cos C — sin B sin C cos a. 

627. sin a ctn & = sin C ctn B + cos a cos C. 

«r.o , i /sin s • sin (s — a) 

628. cosi-^=-V • r, ■ -• 

^ ^ sin b ■ sm c 



««« . , A /sin (s — 6) • sin (s — c) 

629. sinAJ = -V — '^ — ^I — ^^ 

^ ^ sm b ■ sm c 



««o. , . /sin (s — ^) • sin (s — c) 

630. ta.ulA = \ ^ i , . ^ ' 

' ^ sm s • sm (s — a) 



78 TrwIGONOMETRY. 



oQi 1 ^ j cos (S -B)- COS (S-C) 

631. GOS^a=\ ^ : j- r-^-; ^- 

» sin f» . sm f ; 



sin B • sin C 



„r,n ■ 1 — COS S-cos(S — A) 

632. smia = V ■■ — „ • ^ ^• 

■^ \ sin « sm n 



CQQ 4- 1 * / — COSTS' -cos (.S — ^) 

633. tania = V eos(^-^).cos(^'-C) - 

2s = a-{-b-\-c. 2S=A + B + C: 

634. cos \{A-\-B) = ^^—^ '- sm \ C. ■ 

coK 1/^ x)\ sin-|-(a + &) . 

635. cos ^ (J — ^) = T-^^-r ^ sin ^ C. 

sin -^^ c 

636. siniM + 5) = 2^^^^^P^cosia 

^ -^ cos ^ c 

637. sin \{A — B) = V^ cos \ C. 

638. tani(^ + B) = 55ll|^etaia 

639. tan«^-^) = ?|±i|^ctnia 

640. tan |(a + ^') = ^"^ t ^^ 7 f x tan i c. 

^ ^ ^ cos |(^ + 5) ^ 

641. tan ^(^ - b) = ^!" t /^ 7 ^x tan ^c. 

g^2 cos ^(a + b) ^ ctn|C _ 
cos ^(a — b) tan -^ (^ f ^) 



ANTITRIGONOMETRIC FUNCTIONS. 79 

In interpreting equations which involve logarithmic and 
anti-trigonometric functions, it is necessary to remember that 
these functions are multiple valued. To save space the 
formulas on this page and the next are printed in con- 
tracted form. 



643. sin-^a; = cos~^ Vl — x^ = tan~^ 



X 



— sec~^ 



= CSC 

X 



1 - = 2 sin-i [^ - i Vl - a;2]i 



= i sin-i (2 X Vr=T2) ^ 2 tan-» \^ — ^ | 

= -^tan-M ^--^^-J=i7r-cos ^a; 

= 4" TT — sin~^ Vl — x^ = — sin~ ^ (— a;) 

= ctn-i^^^^!— ^^ = (2w-f^)7r-nog(a;+Va;2-l) 

= i TT + ^ sin-\2 a;2 - 1) = ^ cos-'(l - 2 x^). 



VT^-^ 1 

644. cos~^x = sin'^ Vl — x^ = tan~^ = sec""^ - 

X X 



= -^TT — sin~^a; = 2 cos" 
= |-cos-i(2a;2-l) 



■v^ 



= csc~^ — ■ = TT — COS" V— aj) 

Vl - a;2 ^ ^ 



= ctn-i ~ 



Vl 



x* 



i log (a; -f Va;^ — 1) = tt — t log ( V^^ — 1 — a;). 



80 ANTITRIGONOMETRIC FUNCTIONS. 

645. tan~^cc = siri"^ — , = cos~^ , = h sin-^ -— — ; 

X 

= -J-TT — tan~^ - 
x- 

L 2 Vl + x2 J L 2 ViT^ J 

= ^ Un- ,-1^, = 2 tan- [^ll+l^'] 

1 — X^ |_ X J 

= — tan~^ c + tan~^ :; = — tan~^ (— x) 

[_1 — ex J ^ ' 

= i * log TT^ — ; = i * log -. 

646. sin~^ a; ± sin~^ y — sin"^ [cc Vl — if ±y Vl — a;*]. 

647. cos~^ X zb cos~^ y = cos~^ [xy if V(l — x^) (1 — ?/^)]. 

648. tan-i a; ± tan-^ v = tan-^ ^^J- i . 

648. sin~^ a; ± cos~^ y = sin~ ^ [a:/y zt V(l — x^) (1 — ?/*)] 

= COS" ^ [?/ Vl — x^ If a: Vl — y^]. 

650. tan- ^ x ± ctn-^ y - tau- ^ f^^^l = ctii- ' [^^1 

651. log (x + yi) = i log (x^ + y^) + i tan"" ' (jj /x). 



HYPERBOLIC B'UNCTIONS. 81 

B. — Hyperbolic Functions. 

652. sinh a; = ^ (e^ — e^^) = — sinh (— x) = — i sin (ix) 

= (csch x)~^ — 2 tanh ^x -7-(l — tanh^ ^^)- 

653. cosh X = ^ (e^ + e~^) = cosh (— a;) = cos (ix) — (sech ic)~^ 

= (1 + tanh^ ^x)^(l- tanh^ ^ x). 

654. tanh a; = (e^ - e"^) -^ (e^ + e"^) = - tanh (- x) 

= — i tan (tx) = (ctuh x)~^ — sinh a; -^ cosh a;. 

655. cosh xi = cos a;. t*^ z^ X ^ ^»*^V 

656. sinh a;* = ^ sin a;. r57 -^^iX ^ eA-A/^^X 

657. cosh^a; — sinh^a; = 1. 

658. 1 — tanh^x = sech^a?. 

659. 1 — ctnh^a; = — csch'^a;. 

660. sinh (x ±y) = sinh x ■ cosh y ± cosh a; • sinh y. 

661. cosh (x ±y) = cosh a; • cosh ?/ ± sinh a; • sinh y. 

662. tanh (a; zt ?/) = (tanh x ± tanh ?/) ^ (1 rb tanh x • tanh ?/). 

663. sinh (2 a;) = 2 sinh a; cosh x. 

664. cosh (2 a;) = cosh2a;-|-sinh^a; = 2 cosh^a: — l = l+2sinh^a;. 

665. tanh (2 a;) = 2 tanh a; ^ (1 + tanh^a;); 



666. sinh (i a;) = V^ (cosh a; - 1). 



667. cosh (J- x) = Vi (cosh a; + 1). 

668. tanh (i a;) = (cosh x — 1) -i- sinh x = sinh a; -f- (cosh a; + 1). 

669. sinh x -\- sinh y = 2 sinh -^ (a; + ?/) • cosh ^ (x — y). 

670. sinh x — sinh y = 2 cosh ^ (x + 2/) • sinh ^(x — y). 



82 HYPERBOLIC FUNCTIONS. 

671. cosh X + cosh y = 2 cosh ^ {x + y)- cosh ^(x — y). 

672. cosh X — cosh y = 2, sinh ^ {x + y) ■ sinh ^{x — y) 

673. d sinh a; = cosh x ■ dx. 

674. c? cosh X = sinh a; • dx. 

675. c^ tanh x = sech^ a; • dx. 

676. 0? ctnh a; = — csch'^ x • c?a;. 

677. d sech a; = — sech a; • tanh x • c?a;. 

678. d csch a; = — csch x ■ ctnh a; • dx. 

dx 



679. sinh-'a; = log (a; +Va;2 + 1) = J" 



Vx^ + l 



= cosh~^ Va;^ + 1. 



680. cosh- 1 X = log (a; + Vx^ - 1) = J 



Va;2-1 



= sinh-^ Vx^ — 1. 



/rfa 
^-3 



x^ 



/rfx 
dx 



683. sech- ^x = log (^^ + yj^, _ 1^ = - J 

684. csch'^x = log f- + yj^ + 1 ) = "X 



X Vl — x^ 
dx 



685. c? sinh-^x = 

686. c? cosh~^x = 



xVx^ + l 
<^x 



Vl+X^ 

dx 



VV 



687. dta,nh-^x = 



HYPERBOLIC FUNCTIONS. 83 

dx 



688. dGtnh-^x = - 

689. dseGh-^x = - 

690. dcsGh-^x = - 



1-x^ 
dx 



x^-1 
dx 



ic Vl — x^ 
dx 



X 



V^+i 



If m is an integer, 

691. sinh (mTri) = 0. 

692. cosh {miri) = cos mTT = (— 1)*". 

693. tanh (mTri) = 0, 

694. sinh (x + mTri) = (—!)'» sinh x. 

695. cosh (x + mTri) = (— 1)™ cosh (x). 

696. sinh (2 m + 1) ^ Tri = ^ sin (2 m + 1) ^ tt = ± i 

697. cosh (2 m + 1) i T^^ = 0. 

-rr ±X ] = i cosh CC. 






799. cosh (— ±a;j=±:i sinh x. 

700. sinh w = tan gd u. 

701. cosh u = sec gd m. 

702. tanh u = sin gd u. 

703. tanh ^ m = tan i gd w. 

704. u = log tan (i tt + ^ gd u). Tsec a; r/rr = r/d' ' 



a:. 



84 ELLIPTIC FUNCTIONS. 



Elliptic Functions. 

dz r* dO 






V(l - s;2) (1 - A"2^=^ Jo Vl-A-2siii2^ 
where A- <C 1, and x = sin <^, <^ is called the amplitude of u and 
is written am (u, mod A;), or, more simply, am w; x = sin <fi = smc, 

Vl — x^ = cos <^ = en w, Vl — A;^a;^ = A<^ = An r< = dn %, 

Hence, am(0)=0, sn(0)=0, cn(0) = l, dn(0)=l, 
am (— u) = — am u, sn (— u) = — sn u, 

en (— u) = en u, dn (— u) — dn m. 



705. sn2« + cn2« = l. 

706. dn2« + /.-2sn2?f = 1. 

707. dn^it - k-' cn'u = 1 - k' = k'^ 
2 sn ?^ • en tt ■ dn ?( 



708. sn2u = 

709. en 2 M = 



1 — k' sn'* « 

cn^ i( — sn^?/ • dn^ u _ 1 — 2 sii^ ?t + k^ sn*u 
1 - A;2 sn* ?< ~ 1 - A;2 sn* le 



_ 2 sn^ ?< ■ dn^ u _ 2 cn^ ?< 

1 — k^ sn* u 1 — k^ sn* u 

„, » - _ dn^ ri — k' su'^ u ■ cn'^ u 1—2 k^ sn^ ?i + A;^ sn* u 

710. dn 2 w = :j — J = — -^^ 

1 ~ k- sn* i< 1 — A;'' sn* u 

_ . 2 l^ ^v?u-Q,v?u __ 2 dn^ u 

1 — k^ sn* M 1 — k^ sn* ?< 

m\ 1 — en ?/ 1 — dn II. dn ii — en m 



711. sn2( 

712. cn^l 



2 J l+dn« k\l + cnu) k'^ + dn u— k^ en u 
n 4- en u k^ en v — k''^ + dn u 



'u\ _ dn 



+ dn u k^(l + en n) 

7c'- (1 +enM) 



A;'^ + dn w. — k^ en w 



ELLIPTIC FUNCTIONS. 



713. dirl-1- i + dui* ~ /l-2(l + cni*) 



85 



~~ /v'^ 4- dn « — ^'■^ en u 

If, moreover, v = I — , = > 

Jo V(l - z') (1 - /tV) 

714. sn^ u — sn^ y = cn^ v — cn^ ii. 

„.. ^ ^ sn ?6 ■ en V • dn v =h en m • sn y • dn w 

715. sn ill ±v) — —. :-, 5 

^ ^ 1 — k^ sn- u ■ sn'' v 

„, _ , ^ en u • en y zp sn ?( • sn v ■ dn ?? • dn v 

716. en (u ±v) = 7^ — 5 i 

^ ^ 1 — A;'' sn- tc • sn'' y 

= en it • en V rp sn u • sn y • dn (u ± v). 

„,„ , . . dn «-dn y =p />;'^ sn ?i • sn y? -en 7/ • en V 

717. dn {u ±v) = ::; —„ 7, s 

^ ^ 1 — k^ sn^ u ■ sn'' v 

= dn ?< • dn v zp /*;- sn ?< • sn ?; • en (w ± v). 

„,„ , , tn ?f -dn f ± tn w-dn it 

718. tn Cu±v) = - 7 , ^ 

^ ^ 1 ^tu u -tn V • dn u ■ an v 

«,« , , s . V 2 sn it-en vdn V 

719. sn (w + v) -\- sn (?t — v)= r; — 5 5— * 

^ "^ ^ ^ 1 — k^ sn'' « • sn'' v 

/ N 2 sn ?' • en u ■ dn ?t 

720. sn (71 + y;) — sn (u — v) = r^ — 7. r~ ' 

^ ^ ^ -^ 1 — /v'' sn'' ?t • sn'' V 

. . 2 en it • en ?' 

cn(?t + w) + cn(it— y)= :j 1-5 3 ^• 

^ ^ ^ ^ 1 — /r sn- u ■ sn'' v 

„„ _ ^ . , . 2 sn ti, ■ sn ?> • dn u ■ dn i; 

Tad. en (it + iM — en ni ~ v)= 77 — 7, ^ 

^ ^ ^ ^ 1 — A:'' sn'' it • sn'' v 

_,-„ ■, / ^ 1 / N 2 dn ?t • dn i> 
7^ J. dn (u -\- v)+ dn (it — v)= 7-; ; ^ • 



86 ELLIPTIC FUNCTIONS. 

_^ . , , , , , , 2k^ smi-snv -cnu-cnv 
724. dn (ic + w) — dn (71 — v)= .. _ , .^ 



sn^ u ■ sn^ V 



sn^ u — sn'^ V 



725. sn(t. + ^).sn(^^-^;) = ^_^,^^,^^_^^, 



-y 



1 — A:^ sn^ w • sn* v 



1 Fdn^ V -j- k'^ sn^ ?* • cn^ v ~| 
A;^ |_ 1 — A;^ sn^ u • sn^ v J 



726. en (u ■}- v) ■ en (w — v) = rr — i ^ 

en^ w + en^ v 1 _ -1 ^^^ ''^ ' ^^^ ^' + ^'^^ '^ ' ^^^^ ^ 



1 — A;^ sn^ tc-su^v 1 — A;'* sn^ w • sn'' v 

727. dn (u + v)- dn (m — v) 

_ 1 — A;^ sn^ ?i — A;'^ sn^ v -\- k^ sn^ ii ■ sn^ v 
1 — k^ sn^ u ■ sn^ t; 

dn^ n + dn^ t? 



-1. 





1 


-A:^ 


sn^ 


tt • sn^ V 




sn?< 


• dn w 


• en V ± 


sn ?' • dn V • 


en u 




1 


-A;^ 


sn^ 


u • sn^ V 




en?*- 


dnw- 


cnz?- 


dn 


vqzk'^ snu 


• snv 



1 — A;^ sn^ u • sn^ v 

„_ _ , ^ , . sn M • en w • dn v ± sn ?? ■ en v • dn w 

728. sn (u ± v) on {u rp v) 

729. sn (m db v) dn (w qr v) 

730. en(^±z;)dn(^zF^) = -^^ ^ j^ — sn^'t^^sn^.;^ 
-n- .-^ , . x-ir^ , x-i (en V ± snw-dn w)^ 

732. sn (wt, A;) = i sn (w, A;') /en (m, A:'). 

733. en (wt, A;) = 1 /en (m, A;'). 

734. dn {ui, k) = dn (m, k')/cn(it, k'). 



bessel's functions. 87 



D. — Bessel's Functions. 



7oO. t/g (x) — 1 2^ "^ 2^ • 4^ 2^ ■ 4^ • 6^ 

736. iq, (a;) = ^0 (a;) • log a; + 2-2 - 2F:4r2 + 22.42.62 



7i! A (- i)^-^.» + 2t 



[a^.= l + i + i + ... + l/k:] 
[When 11 is an integer 



737 T I't\ — "V 5^ -^ [When ?i is an integ 

• « I . t/„ ,^x; r (?i + 1) ^ 2" ■^'^^'■kl(n + k) ! ^19 may be used.] 

738. lK(^)=Jn(x)-logx-^JX ^''~2^^.iy''' 







739. According as n is or is not an integer, A ■ J^i^) + B ■ K„(x), 

or A ■ J^(x) + B ■ J_ ,^(x) is a particular solution of Bessel's 

equation, fp.^ ^ r?^ / w^ 

+ -■—+ 1--Az = 0. 



740. f/Jg (a;) /riic = — J^ (cc) ; 6? [ic" ■ J"„ (x) ] /(7ic = x" ■J^_i (x) , 

if /i > I ; (^[a;-« ■ J,Xx)ydx = — x.-" ■ J"„+i(a'), if ?i > - 1 

741. J,^_,(x) - J„^,(x) = 2 . dj„(x)/dx ; 
2 n ■ J^(x) = X ■ J"„_i (x) + x- J„+i(x). 

When x is large it is sometimes convenient to compute 
approximate numerical values of J^ (x) by means of the semi- 
convergent series. 



.^xrt x / N 2 r„ r (271 + 1)77 1 

742. ^„(^)=^— I^P^.cosj^^ ^^_^| 



(4 n? - 1) (4 n" - 9) 



^ . f(2n + l)7r 



-.}]. 



(4 rv" - 1) (4 ?i^ - 9) (4 v? - 25) (4 rv" - 49) 
^ 4 ! (8 .x)* 

744 n - ^^'-^ _ (4^^-l)(4r.'^-9)(4n^-25) 

'"~ 8x 3! (8^)^ "^ 



88 



SERIES. 



E. — Series and Products. 



[The expression in brackets attached to an infinite series shows values 
of the variable which lie within the interval of convergence. If a series 
is convergent for all finite values of x, the expression [x^ < co] is used.] 

745. (a + by = a" + na^'-'^b 

, n(n — V) „,„ , , n\ a^~^b'' , ^,0^0-, 

+ 2! "''+•• • + (,.- A)! ,!.! +• -•■[*<"■] 

746. (<i-Sx)-' = iri+- + ^ + ^' + ---1- pV<a».] 

Ct I Cv ct- (M I 

747. (1 zb xy = l±nx-\- ^ ^^~ ^ x^ 



2! 
n{n — V) {n — 2) x^ (± 1)^' n\ x^ 

" 3] H • • • + ■^^^TT^yr^ + 

748. (l±a;)-» = l=F?^^ + ^^^^^Vr^^' 

2! 



[a;2<l.] 



3! ■ "^ ^^-^ {n-iy.k\ 



749. (l±c.)i-l±ix-^.T^±|^a;^ 



[x^ < 1, 



J 






750. (l±a,)-l = l=pj^+l^,x«^l^a^ 






751. (l±.). = l±i.-y.'±if|^^ 

1-2.5.8 , 
3-6 -9 12 "■ 






lx'<-L] 



SERIES. 



89 



1-4 , 1-4-7 , 
752. (l±a:)-§ = l + ix + g^x2rF3-^a;' 



1-4. 7-10 , r "^1 1 

H 7i cc* zp • • •• \x- < 1.1 

^3 -6 -9 -12 ^ •- ^ 

, a;* 1.3a;« 1.3-5ic« 

753. (i±^¥ = i±i«^^-2:4±2:4:6~2T:6:8-'"- 

[a;2<l.] 
, 1-3 , 1.3-5 , , 

754. (l±^')-^ = l + i^' + ^^ +27176 "^^ ■■■ 

[x" < 1.] 

755. (1 ± x)-' = l^x + x^-px'' + x*zpx'-\ . [a;2 < 1.] 

31 , 311 3 

756. (l±x)^ = l±ix-h^^x-:+j-^-^x' 

3-1-1-3 3.1.1-3.5 ,<.^-j 

+ 2.4.6-8 ^2.4.6.8.10 ^ •- ^ 

757. (l±^)-? = l + l^ + 2T4^'^t7I76^'"^"*' t^""'^^-^ 

758. (I±x)-^ = 1^2x + 3x''zp4:x^ + 5x*^6x^+ ' • : 

Ix' < 1.] 

759. e- = l+x + |-' + |-j + ---. [a;=^<co.] 

Cicloga)^ , (x log ay , r 2^.^ n 

760. a- = 1 + X log a 4- '^ ^° ^ + ^—o, + ' " •• [^'<^^-] 

761. i(«^ + e-^)=H-| + fi + fi + - •• [^^<^-] 

762. i(e^-0 = a^ + fj + fj + f-! + '' ^- [x2<=^.] 

763. «-" = l-^^ + fi-fi + li • I^^<oo.] 



90 SERIES. 

A series of numbers, B^, B^, B^ •• •, of odd and even 
orders, which appear in the developments of many functions, 
may be computed by means of the equations, 



2 n (2 n - 1) 

2! 



-^2n o, J^2n — 2 



2n(27z-l)(2n-2)(2n-3) _ _ 

—^ ^-B,„_,=(2n-1)B,,_, 

_ (2.-l)(2.-2)(2.-3) ^^^_^^...^_^^„_,^^^ 

Whence B, = h^2 = 1, B, = ^l, b, = 5,B, = ^\, B^ = 61, 
B, = ^L, B, = 1385, B, = /^, Ao = 50521, Bn = ^Wo. ^12 = 
2702765, Bis = i, etc. The ^'s of odd orders are called 
Bernoulli's jSTumbers ; those of even orders, Euler's Numbers. 
What are here denoted by -Z>2n-i ^nd Bz,, are sometimes rep- 
resented by B„ and £J„, respectively, 



7? 9 



'2n — 1 



(2?i)! (22«-l)7r2« 
(2n 






02n + 2r -I 1 1 1 



7fi4 ^ _^ rr Ax^ B,x* B,x' B,x' 

e^-1 2 2! 4! 6! 8! 



[a;<2 7r.] 



765. log X = (x - 1) - i(x - ly + i(x - ly . 

[2>a;>0.] 






[^>i-] 



SERIES. 91 

[a^>0.] 

768. log(l-{-x) = x-ix^ + ix'-ix* + -- -. [a;2<l.] 

769. \ogCj^^ = 2[x + ix'-b^x' + \x'+-- 'I [rr2<l.] 

771. log(a;+Vl+a;^) = a;-— + ^^j7g- ^^ g^ + -- •. 

[a;2<l.] 

Series for denary and other logarithms can be obtained 
from the foregoing developments by aid of the equations, 

log„a; = log, a: • log„e, log.x = log„a; • log, a, 
log, (—z) = (2n + 1) iri + log,s. 



772. sin £c = a; - |-j + |j - li H . [x^<co,] 

jp2 ^4 ^6 

773. cosa; = l— 77T + — — TTiH = 1 — versinic. [x^ < oo.] 

2! 4! 6! 

774. tan a; - a; + g + ^g + g^g + 2835 

02n/02»i 1\ J> rp^n — l 

+ •••+ (2.)!"" +•••• [^^<i-^-] 

^py_ J^ «// JO ^ (/^ t// 

775. ctna; = ----^- — -^^ 

a;(2?i)! •- -■ 



92 SEEIES. 



776. sec. = l + - + — +— +.- + -^^ + -.|_.^<-J 

777. csc. = j + - + 3-g-, + — 

ftmo • -1 i"'^ L • o X i. ■ o • O X 

778. sm>a. = x + - + 2:^.- + 2:^.- 

+ ■ • • = ^ TT — cos~^a;. [a;^< 1.] 

779. tan-^cc = x — ^x"^ + ^x^ - \x'' + • • • = ^17 — ctn-^x. 

[a;2<l.] 

780. tan-ix = ^-i + -^-p^3+-.-. [a;^>l.] 

2 a; 3 a;'' 5ic^ ■- -* 

„oi 1 ttI 1 1-3 1.3-5 

781. sec~^a; = — — 



2 a; 6a;' 2.4.5a;^ 2 • 4 • 6 • 7 a;' 

= ^TT — csc"~^a;. [a;^>l.] 

782. log sin a; = log a; - i a;^ - ^\^ x^ - ^^l^ a;« 

22"-ii?,„_ia;2» 



?i 



(2^0! 



[a;2<7r2.] 



783. log cos a; = - -^ a;2 - J^ x^ - J3. a;« - ^\\^ x» 

027J-1 /92n _ -|\ 7? ^2n 

?i (2 /«) ! L * J 

784. log tan a; = log x + ^ a;- + /^ *"* + ^§§5 ^^ 

/92n-l _ -|\ 02« » „2n 

w(2w)! L -i J 



•yft*; Bin. 1 . ,^' Sa-" %x^ 2,x^ mx' ^ 



[x2 < 00.] 



SERIES. 93 



786. e-- = e(^l-- + — --g^+---J- [a:^<cc.] 

787. e-^ = 1 + ^ + ^ + ^V ^' + ^ + • • •. [:.^<i7r^] 

788. .sin-^ = l+:. + | + ^ + ^4----. [x^<l.] 

789. e'^""'" = l+^ + |'-f -||-' • [^^<1.J 



/yt'i /ytb /yti 

*Aj tAj \Aj 



790. sinha; = a; + - + gj + y^H . [a;=^<^.] 



rg\^ /yt^ /yiO ™0 

»A^ «Ay »A.' »Ay 



791. coshx = l + - + - + - + -+---. [a;2<Q0.] 



2! 4! 6! 8! 



£c .^. ^. ^A ^ X 



3 



792. tanh x = (2' — 1) 2^^! — - (2* - 1) 2*^3 t] H 

= :S[(-l)"-^22«(22»-l)^2„-ia;'"-V(2w)!]. 

793. ctnh ic = - (1 + 2 [(- 1)"-' 22» j52„_i cc2V(2 ti) !]). 

[a;2<7r2.] 

794. sech cc = 1 + 2 [(- 1)» ^2„ ic'V(2 »0 H- [^' < i t^'-] 

795. csch X = - - (2 - 1)2 B,^^ + (2^ - 1)2 B,^ 

= i (1 + 2 2[(-l)"(2^"-^-l) ^2„_x 0^27(2 ^On)- 

[a;2<7r2.] 

796. sinh-^ = x-^x^ + ^^'-|^;|^ + --..[:«^<l.] 



94 



SERIES. 



797. tanh-ia; = x + | + ^ + ^+- • •. [a;^<l.] 

798. ctnh-i ^ = --^T-s + ^5+-"' [a;2> 1.] 

7QQ 1.-1 .1 1 , 1-3 1-3-5 

7yy. cscn x-^ 2.3.x3'^2.4 Sx^ 2.4.6.7 •x'-''^ "'• 

800. J|^V-'(Zic =a; - ^ x^ + ^ - -^^ + • • •. [a;2< QO.] 



801. 



n5 /yt9 /j^lo 



J^'COS (X') <to = 0! - g^, + g^ - j^, + • ■ • . [^» < «.] 



'•Xt 



802. I ^— ^ = - ^ + 



+ x'' a a + b a + 2b a + 3b 



+ • • •. 



803. f(x + h) = f(x) + h •/' (x + Oh). 



h' 



804. f(x + h) = f(x) + A ./' (X) + - /" (x) 



805. /(x -h A) =f(x) -{-h-f'(x) + -/"(x) 



nl ^ ' 



+ 



A" 



+ (;^^^-^^~^^""'--^"^'^ + ^^> 



806. /(x + A, y + A;) =/(x, li) + 7i/'^(a; + ^7i, y 4 ^A;) 

+ ^/'^(x + ^A, y + ^A;). 



807. 






SERIES. 96 

i (,/2ii^ + 3 « ^£(^ + 3 M' m^ 



3!V ^^^ ^ ^Z/-^-^' ^-^-^y 



+j,M^y...^E^ 



■.f(x, y) + QiD^ + ^^.)/(^, y) + I; {hD. + A-i),)y (a;, 2/) 

ft • 



««« . 4 r . TTCc , , . Sttcc , , . Stto; n 

808. 1 = - sm h i sin 1- i sm + • • • • 

[0<ic<c.] 

o«« 2 (■ r . Tra; , . 2 ttsc , , . 3 ttcc ~| 

809. x = — sin A^ sm h ^ sm • ■ • 

TT [_ c c c J 

l-c<x<c.'] 

c ^^-r TTiP , 1 3 7rar 1 ^irx ~| 

810. ^ = ___^cos- + 3,cos— + ^cos-^ + ---J- 

[0 < a- < c.-] 
2'irx 



«,-. . 2rr/7r2 4\ . irx tt^ . 
811- -^=^LVT"l/"~~"2'^ 

/tt^ 4\ . 37ra; tt^ . 4 
+ U-3-3>^"-^-4^^"- 



+ (—--, )sin — + ••• • [0 <»•<('.] 

2 TTir . 1 3 irx 



5 5V c 



C^ 4 C^ r TTX 1 



812. a;"-^ = r cos ^ cos h ^ cos 



32 6- 

42 — c 



_l,cosi^ + ---]- l-c<x<c.-] 



m 



SERIES AND PRODUCTS. 



813 log sin ^x = — log 2 — cos x — ^ cos 2 a; — -^ cos 3x — • ■ '. 

[0<a;<^7r.] 

814. log cos ^x = — log 2 + cos x — ^ cos 2x + -^ cos 3 .r — • • • . 

[0<ir<i7r.] 

815. /(x) = i ^0 -I" ^1 cos — - -\- b^ cos (- • • • 

. Tra; . 2 Tra- ^ ^ -, 

+ «! sin h agsm 1- • • •, f— c<.x<.c.\ 

c c •- -" 



where ^,„ = - I / (a) cos da, 

1 r+'^ 

C •>' — c 



?ft7ra , 
sm da. 



816. sin^=^ 1 



[' 



W,L 



817. cos^ 



.27r. 



1-1 



2^ 



=[-(i')'] [-©)■] [-(.")•] 



2^-4^6^ •• • (2m)^(2?>i + 2) TT 
12.32.52 ... (^2 m + 1)-' 2 

2^ -4' -6" • • • (2my(2m + l) 



12.32.52 • • • (2m + ly 



819. 



^"^^~2"n!l 2(2/^ + 2)^2- 



a" 



(2 /i + 2) 2 • 4 (2 ?? + 2) (2 7i + 4) 



.r 



2 • 4 ■ 6 (2 ;i -h 2) (2 w + 4) (2 ft + 6) 



TT + 



} 



820. 



DERIVATIVES. 97 

F. — Dekivatives. 

d (au) a du 
dx dx 



R91 d{u + v) _du dv 
dx dx dx 

833. -^^; — - = V - — \- u--' 
dx dx dx 



fu\ du dv 

QOQ \'V _ dx dx 

dx v^ 

824 ^/('^) = df(u) ^ du ^ 
dx du dx 

d\f{u) _ df (Ihi (lf_ dtl 
dx^ du dx^ du? dx^ 

826. ^ = ?^a;«-^ 
dx 

827. ^ = e-. 

dx 

oo« -^*" du . 

829. '^^x^{l^\o^,x). 

d{\o^^x) 1 log„e 



830. 



o?iC x • logg a a; 



_-, c? sm a; 

831. — ; = cos X. 

dx 

d cos X 

833. — ; = — sm X. 

dx 



98 DERIVATIVES. 



833. — % = sec^ic. 

ax 



834. — = — csc^ic. 

ax 



835. — ' = tan x • sec x. 

ax 

836. — ; — = — ctn x ■ CSC x. 

ax 

««-, (I sin~^a; 1 

837. ; -- 

ax 

(I cos~^a:; 
dx 

d tan~^aj 
dx 

d etn^^.-B 
dx 

d sec~-^a; 
dx 

d csc~'rK 



838. 

839. 
840. 
841. 
842. 



Vl-x^ 


-1 


Vl-x« 


1 


1+x' 


1 


l+x2 


1 


x Vcc^ — 1 


1 



-.„ d^rahx . 

843. ; = cosh a;. 

dx 

_. - (^ cosh a; . , " 

844. — ^- — = smh x. 

ax 

o._ d tanh x . „ 

845. ; = sech^ x. 

dx 

846. ^li^=-csch^... 

dx 



DERIVATIVES. 99 



847. = — seen x ■ tanh x. 

ax 

848. = — cscli X ■ ctnli x. 

ax 

_._ d sinh~^a:; 1 

849. - 



dx -yjx^ + 1 

d cosh" ^ a; 1 



850. 
851. 
852. 
853. 
854. 
855. ^^£j{x)dx = f{h) 



dx -^x'-l 

d tanh~ ^x _ 1 
dx 1 — x^ 

d ctnh^^a:; _ 1 
d sech~^a; — 1 



dx X Vl — x^ 

d csch~^a; — 1 



dx X Va;"' + 1 

d ^^ 



db. 



856. ^fy(x)dx = -f(a). 

857. jjjix, c) dx =£l)J(x,c) . dx +f(b, c) g - f(a, c) ^• 

r.,ro d«(u-v) d"u , dv d"-'^u 

858. — ^^ = V ■ h n-- ; r 

dx" dx" dx dx"-^ 

n(n — V) d^v d"-^ic , d'^v 

^ 2! dx^ dx^-^ dx'' 

859. If f(x, y, z, • • •) is a homogeneous function of the wth 
order, so that /(Ax, Xy, \z, • • °) ^ X"/(x, y, z, • • •), 

x-DJ+y.DJ+z.DJ+-'- = 7if. 



100 DERIVATIVES. 

860. Ux = <f>(y), 

dy _ 1 A _ _ < ^"(y) 



15 



861. If sc = f{t) and ?/ = (^ {t), 
dy^£({) d'y_ f'(t)-<f>"(t)-f"(t).<f>'(t) 

dx f'(ty dx' [f'(t)Y 

862. Uf(x,y) = 0, 

dy ^ V >^f_ DJ 
dx dx ' dy D^f 

dhi _ D,y • ipjY - 2 D^BJ. DJ. DJ+ D,y ■ jDJ) 
dx' {DJ) 



863. If y =f(u, v), u = <f>{x), and v = \p{x), 

d£ ^^i dM di <ii^ 

cZx gw dx ^ dv dx ^«/ + ^ ^"/' 



^_ay /^iA" ay du dv_ dy^ 

dx^ du^ \dxj du ■ do dx dx d'' 

df d?u df dh 
du dx^ dv dx^ 



■v \dxj 



= u'' ■ D\f +2u'-v'- L„ DJ+ v'^ ■ DJ^f 
+ u".I)J+v".DJ. 

864. If f{x, y, z) = 0, D^--^- DJjDJ, 
DJz = -lD:'f.{DJf 

- 2 DJ. DJ- DJ)J^ D,y{DJy^J{DJf, 
Djy^z = - ID^DJ- {Djy - DJDJ. D^DJ 

+ DJ. DJ. DJ)J+ DJ. DJ. DJ^Iipjy 



DERIVATIVES. 101 

865. If r=</)(w, v), u=fx{x,y), and v=f^{x, u), 

D^ V= d: <i> . (D,uy+ d: </, . {D^vf + 2 D^D, ^ . D,u ■ D,v 
+ 2 i>„D,«/> • {_D,u ■ D^v + D,jU ■ D.y] 



In the special case, u^r = Va-'^ + y'^, v = 6 = tan~^ (^//a-), 
we have D^x = cos = x/ Va;'^ + ?/- ; D^y = sin = y I Va;- + ?/^; 
Z>0X = — r sin = — y \ D^y = r cos ^ = a; ; 



D^r = a; / Va;^ + y'^ = cos ^ ; Z>^r = ?// •va^M-l? = sin ^j 

I^:P=-y I (^' + 2/') = - sin ^/r- ; 
Dyd=x / (x^ + y^) = C0Bd /r\ and 

Z>/ F + X*,;- F = Z>,^ F + - -i), V+\- De" V. 

866. If F= «^(?<, v)' ''*=/iO^ ^). and v = f^{r, 6), 

2>,2 F + ^ • A- F + i ■ A' F = i)„2 F- [(Aw)' + ^^$^~\ 



r r 






102 DERIVATIVES. 

867. If V=4>{u, V, iv), u =fi{x, ij, z), V =f^{x, y, z), and 

w=fs(x, y, z), 

I)JV= BJ^V. (R,icy + D^'V. (D^vf + DJV- (D^wf 

B^ V + D; V + D,'V= D^'V- [{D^uy + (!>,«)'+ (A^O'l 

+ i),^ r[(D,^(;)^ + (D^tvy + (D^wy^ 
+ 2 i)„A F- [ J>,tt . X>,^; + D,it ■ J>,v + JD^u ■ A«] 
+ 2 1),A,V- IB.v ■ D^w + D,^v . D,jw + D,v ■ B.w^ 
+ 2 i)„.i)„ V- [Bjv ■ JDji + D,/a ■ D^u + D^^v ■ i),w] 
+ D„F.[i)> + I>/« + A'^*] 

In particular, if 

a; = r sin 6 cos <^, y = ?' sin 6 sin ^, z = r cos ^, 
30 that M = j-^^ = a;2 + y2 _^ ,-s^ i; - ^ = tan-^ ( Va;^ -f- f/z), 
w~^~ tan~^ {y /x), we have 

Z>,.« = cos ^ = s/ Vx^ + ?/ + .v^ ; 
D^x = sin ^ cos ^ = x / VxM- y^ + ^^ ; 



DERIVATIVES. 



103 



r^ sin 6 



V.y — sin 6 sin <^ = y I V.^'- + if + z^\ 

I)qZ = — r sin ^ = — Vcc^ + if ; 

jD^ = r eos ^ cos <^ = zx / ^j^ + y^ : 

J)^y z= r cos ^ sin ^ = zi/ / Vo;- + 3/^ ; 

B^z^O; 

D^x = — r sin ^ sin ^ = — y ; 

X)^y = r sin ^ cos <^ = x ^ 

7)_r = s/r = cos 6\ 

D£ = - Vx-2 + //r^ = - sin 0/r j 

Z)^?- = X /r = sin ^ cos <^ ; 

J)^0 = xz/7-^ 'Vx^ -{- f = COS cos <;?>/r; 

^;«<^= -I//(^^ + f)= -sin<^/r sin^j 

-^/ = y/-'" = sin ^ sin <^ ; 

Z)y$ = ^1/ / i^ Va;^ + y"^ — cos ^ sin ^/r; 

X>j,<^ = X / (x^ + U') = cos ^ /r sin 6 ; 

(i>,r)^ + (i>,r)^ + (D^rf = 1 ; 

(DAY + (A,<^)^ + (A<^)' - 1 /'-^ siii'^ ; 

D^^V + I),fV + B.^V 

D,{r- ■ B, V) ■sinO + ^^ + A(sin 6 ■ A F) 



104 



DERIVATIVES. 



868. If X =fi{u, v), y =/2(m, v), z =fz{u, v), 



D^ = 



_ D..f,-Dj\~DJ,-D,.f, 

869. If X =/(«, u), and y = ^(z, u), 

Byz = BJlip,^. BJ- BJ. i>„<^). 

870. If F^ (x, y, z, u, v) = 0, 

F^ (x, y, z, u, v) — 0, and F^ (x, y, z, u, v) = 0, 



B^ 



B,F, B^F, B^F, 
B^F, B^F, B,F, 
B,F, B^F, B,Fz 



B,F, B^F, B^F, 
B,F, B^F, B,F, 
B,F, B^F, B,F, 



871. If F^ (x, y, z) = 0, and F^ (x, y, z) = 0, 

cly dz 



B,F, . B,F^ - B,F, . B,F, BJ\ • B,^F, - B,F^ ■ B,,F, 

dx 
ByF^.B,F^-B^F^-B,F,' 



If each of the quantities y^, y^, yz, • • • 2/„ is a function of 
the n variables x^i x^, x^, ' • • x^, the determinant, 

B^^yi B^^j^ B^^y^ • • • 
B^^y^ B^^j^ B^^j^ ■ ' • 



B^^y„ B^,^y„ D^ij,^ ■ • • B^j„ 



873. 



DERIVATIVES. 105 

is called the functional determinant or the Jacobian of the 
^s with respect to the a;'s and is denoted by the expression, 

g,jr2 g(yi> y2> 2/3, •• • Vn) . g (^^^1, ^2, X^, • ' ' X„) ^ ^ 
d {Xi, a-2, Xs, ' • ■ Xn) d (3/1, y2, Vz, • • • Vn) ~ 

d (Vl, ?/2, Vs, ■ ■ • Vn) . g (^1, ^2, ^3, • • • ^„) 
g (^Ij ^2) ^3j ■ ■ ' ^n) g (^1? "^25 X^, • • • X^ 

^ d (]/l, 3/2, y?., ■ • • Vn) 
(Xi, X^f Xg, • • • X^) 

If the ?/'s are not all independent but are connected by an 
equation of the form (f> (jji, ?/2, ys, ' ' ■ y„) = 0, the Jacobian 
of the ?/'s with respect to the cc's vanishes identically ; and, 
conversely, if the Jacobian vanishes identically, the ?/'s are 
connected by one or more relations of the above-mentioned 
form. 

The directional derivative of any scalar point function, u, 
at any point, P, in any fixed direction PQ\ is the limit, as 
PQ approaches zero, of the ratio of «q — Up to PQ, where 
^ is a point on the straight line PQ' between P and ^'. The 
gradie7it, h^, of the function ti at P is the directional deriva- 
tive of M at P taken in the direction in which w increases 
most rapidly. This direction is normal to the surface of 
constant m which passes through P. 

874. K' = {D,u)' + {D^n)' + {D^uf. 

The directional derivative of any scalar point function at 
any point in any given direction is evidently equal to the 
product of the gradient and the cosine of the angle between 
the given direction and that in which the function increases 
most rapidly. 



106 MISCELLANEOUS FORMULAS. 

The normal derivative, at any point, P, of a point function 
u, taken with respect to another point function v, is the limit 
as P(^ approaches zero of the ratio of «q — tip to Vq — Vp, 
where ^ is a point so chosen on the normal at P of the 
surface of constant v which passes through P, that Vq — Vp 
is positive. If (u, v) denotes the angle between the directions 
in which u and v increase most rapidly, the normal derivatives 
of u with respect to v, and of v with respect to u may be 
written 

h^^ cos (?<, v) -7- 7ij,, and A„ • cos (w, v) -v- ^„ 

respectively. If A„ = h^, these derivatives are equal. 

Gr. — Miscellaneous Formulas. 

If s is a plane analytic closed curve, n its normal drawn 
from within outwards, and dA the element of plane area 
within s, the usual integral transformation formulas for the 
functions u and v which, with their derivatives of the first 
order, are continuous everywhere within s, may be written — 

875. I M • cos (x, n) ds = \ i D^u ■ dA. 

876. j [w • cos (x, 7i) + V ■ cos (?/, w)] ds=^ C C(D^ti + DyV) dA. 
Sn. Jb„u .ds= C C (B/u + Dyhi) dA. 

878. j'^iD.n . Djj + D,^u ■ D^v) dA 

= Ju ■ D^v -ds- C Cu (Z)> + D,fv) dA 
= Cv . D^u .ds-^ifv {B^u + D,fv) dA. 

879. f C\ {D^u ■ Djo + D,;ii ■ IJ,/-) dA = Cxu- D„v ■ ds 

-ff ' U'. (^ • A'O + ^. (^ • ^>M dA 



MISCELLANEOUS FORMULAS. 107 

If ^ and 7} are two analytic functions which define a set of 
orthogonal curvilinear coordinates, and if (^, n) and {-q, n) 
represent the angles between n and the directions in which 
^ and 7], respectively, increase most rapidly. 

880. ^j'h^ ■ \ • A ( r ) ^^ =X" ■ ^°^ ^'^' ^^^ ^^' 

881. ^ ^ h^ h^-DA^jdA =fu . cos (i, n) ds. 

882. If r is the distance from a fixed point, Q, in the coordi- 
nate plane, 

/cos (v 71) cl'S 
— '-^^^ — —— = 0, TT, or 2 TT, according as Q is without, 

on, or within s. 



If a9 is an analytic closed surface, n its normal drawn from 
within outwards, and dr the element of volume shut in by S, 
the usual integral transformation formulas may be written — 

883. r Cu cos (x, 71) dS= C C C D^u ■ dr. 

884. I I [y« cos {x, n) + v cos (y, n) + tv cos (z, n)"] dS 

= f f fi^x'if' + I>y^ + D,w)dT. 

885. r rz)„« • (7s = ( ( r (^/« + ^2,''* + a'«) <^t. 

886. j" ^ j" (7),^« . D^v + i>^7f . D^v + i),-a . X>,y) dr 

= r r« -D^v-dS- C C Cu (Bj'v + Z>/y + n^^) dr 
= f f^- ^nU dS- C C C r {Dju + Dfa + D.hi) dr. 



108 MISCELLANEOUS FOKMULAS. 

887. fff>^ {D^u ■ D,v + DyU ■ D^v + D,u ■ D,v) dr 

- ^ ^ ^ vlB^iXB^u) + D^iXD^u) + D,{XD,u)^dr, 

If I, rj, I are three analytic functions which define a system 
of orthogonal curvilinear coordinates, 

889. jyj"/'| • ^ • h^ ■ Dr, (j;^) ^^ =ff"' ■ cos (^7, ?0 ^'S^- 

890. ////^f • hr, .\-D^ {irrh) '^^ =//" • ^°' (^' '') '^'^- 

891. If r is the distance from a fixed point, Q, 

/cos ^?' ??'^ 
-j—^ dS = 0, 2 TT, or 4 TT according as Q is without, 

on, or within S. 

Stokes's Theorem, — The line integral, taken around a 
closed curve, of the tangential component of a vector point 
function, is equal to the surface integral, taken over a surface 
bounded by the curve, of the normal component of the curl of 
the vector, the direction of integration around the curve form- 
ing a right-handed screw rotation about the normals. 

If X, Y, Z are the components of the vector, 

892. C{Xdx + Ydy + Zdz) = C C[(D,,Z - I), Y) cos (x, n) 

+ (I),X ~ D,Z) cos (t/, n) 

+ (B, Y - DyX) cos {z, n)-] dS. 



MISCELLANEOUS FORMULAS. 109 

Equations 893 to 897 give Poisson's Equation in orthogonal 
Cartesian, in cylindrical, in spherical, and in orthogonal curvi 
linear coordinates. 



893. v2r=Z»/r+X>/F+ A'^=-4 7rp. 

1 



894. ~I),(r.D,V) + ^-De'r+l),^V=-4.7rp. 



895. sme.DJr^-D,.V) + ^^ 

^ ^ sm 6 

+ Dg (sin ei)0V) = - A Trpr'' sin 6. 

896. 7i/ ■D^^V+ /i,2 ■D^W+ hi ■ Dl V 



397. ;,,./,,. /.,{i),(,^^^.A^')+ A 



_rL 



•i>„F 



y.,A, ^ 



H. — Certain Constants. 

7r = 3.14159 26535 89793 
logio7r = 0.49714 98726 94134 

- = 0.31830 98861 83791 

TT 

TT^ = 9.86960 44010 89359 

V^ = 1.77245 38509 05516 

logio 2 = 0.30102 99956 63981 

e = 2.71828 18284 59045 

logio e = 0.43429 44819 03252 

log, 10 = 2.30258 50929 94046 

log^2 = 0.69314 71805 59945 

togiologio e = 9.63778 43113 00537 

log^7r = 1.14472 98858 49400 



+ ^dj±-^<r)^ = -*^P 



110 FORMULAS OF INTEGEATION. 

I. — General Fokmulas of Integration. 

F and / represent functions of x, and F\ /', F'\ f", their 
first and second derivatives with respect to x. 



898 
899 
900 
901, 



Cf' •/• dx = Ff- Cf-/' ■ dx. 

. C{Fy-F'-dx = (Fy+^/(u + i). 

C(aF + by ■ F' ■ dx = (aF + by + ^/a (n + 1\ 

J(F +fy -dx ^Jf(f +fy-'dx +Jf{F +f)-'dx. 



902. CF'/(Fy-dx = -l/(n-l)(Fy-\ Cf'/F- dx = logF. 

903. /(^'-Z- F.f ')/(/)''. dx = F/f. 



. f{F'-f- F-f)IFf. dx = log {FIf). 

J dx _ _1^ r dx d_ r dx 

F-{x^ -a?) ~2^,J F-{x-a) ~ 2~aJ F ■ {x + a) 

r dx _ r dx r dx 

■JF(F±f)~' J F.f^J f{F±f)' 



/pi fly, 

, = (2^aF+b)/a. 

y/aF + b 



C F'-dx , ,„ /-— , 

I , = log {F + -^I^' + a). 

/ Fdx _ a r dx b f dx 

{F+a)(F + b) ~ a-bJ Y+~a. ~ a-bJ F+b 

r F-dx _ r dx r fdx 

J (F+fY~ J (F+ f)"-' ^ J TF 



910 



{F+fy j(F+fy-^ ^{F+fy 

911 r_Zji^ = l -I'lZ r F'-dx ^ 1 ^ qF-p 



FOEMULAS OF INTEGllATION. Ill 



913. f f! '"". =- tan-', 
J F^ -\- a} a \ a 



F'-dx 1. ^lF\ 

aF — b 

aF +b 



914. I -—- — T7, = TTT ^°S 

rF^^dx _ r F^-dx r t^"-. 



or./; 

{F'^ -h) V 2(i'^»+^' 






F'^d^__ 1 

+ hF~ h '"^ aF +b 



917. " .. . , „ = T loR- 






919. f , ^' = y sec- ' ( ^ 



r 



J — INTEGRALS Useful in the Theory of Alternating 

Currents. 



922 
923 

924 



I sill ((nt -{- (f>)dt = • cos (wt + </)). 

. I cos (oit -\- ^)dt = -- sin (oit -\- <(>). 

/I 1 

sm^(u)t + <f)dt^-t — -— sin 2 (i^t + </>). 



112 AUXLLIAIIY FORMULAS. 

925. / sin (uyf + <^) ■ cos (wf + cl>)dt = — - sm'^(wt + <}>). 

/I 1 

cos2(o)^ -\- <t>)dt = -t + -— sin 2 (u)t + cfi). 



927. fsin (.ot -\- X) ■ sin (a>^ + /x) dt = ^^^ — ^ (o)^) 



sin (<i)t + A) • cos (wt + /^) 
2 (1) 



_„_ I . , , , ^ , sin (<Dt + A) • sin (wt -\- a) 
928. / sin (oyt + A^ • cos (oyt 4- m) dt = ^^ — ^^-^ ^^ — ^^-^ 



/ sin ((at + A) • cos (wt 4- /u.) f/i^ 

sin (/Lt — A) 



(-0- 



929. / cos (wt + A) • cos (wt + /x) fZi = ^"^ ^^ ^ (w^) 



sin (<«>^ + A) • cos (wt + A) 
2 w 



^„^ C . , . , ^ , sin r???f — n?' + A — Atl 

930. / sm (»^ ^ -I- A) • sm (ni + yu) </^ = ^-—-^ =4 ^ 

J 2 (?« — n) 



sin [w/ + ?t^ 4- A + /"■] 
2 (m + w) 

931. f cos (mt + A) • cos int + ix) dt = — ~- ; — ; 

J ' 2 (/?» + n) 

sm\^nit — nt -{- X — fi'] 
2 (in — ri) 

o,«« r ■ , s , s , cos r??2i + nt + X+ ul} 

932. / sin r^/^i + A) • cos ^nr + n) dt = ^-— 9= , ^ ^-' 

2 (m + ?'') 

cos [?wi — nt -\- X — fJi] 

2 (»i - n) 



J. I sin (?//i -}- A) • cos (ni + fx) dt = 



AUXILIARY FOIlxMULAS. 



113 



933 



'. I cos ((tit + A + vi.r) ■ COS (oj^ + A — III J') dx 
= cos'^(w^ + A) 



inx -\- sin m.r • cos iiix 
mx — sin mx • cos inx 



2«i. 



-sin2(w;' + A) 

m- sin(w/'4-<^)4-?i- cos(w)'+<^) = V»r+H.2- sin(a)f+<^+c)^ 

\ where tan c = n/m. 

III ■ sin(<Df+^)— w • cos(w!'+<^)= V//r+?i-^ • sm{(t)t-\-<i>—d). 



934. 



/ 



e^-^^"^'(h^ 



-h^ cl 
W + c^ 



^e 



.(-b±ci)t 



-ht 



= 77 ; [(c ■ sin ct — b- cos ct) =F «' G> ■ sin r?' 4- c • cos cf)! 

0^ 4-c^^ 



r.-ht 



—= [sin (cf - 8) =F '■ • cos (c;^ - 8)] 



935. j e"' ■ cos (<of + tf>) (If 



a^ + o, 



where tan S = /^/c. 



^ [to sin ((of + ^) 4- « • cos ((of + <^)] 



= — -==:^ cos [to;' + </) — tan ^(a)/«)]. 
936. C (>"'■ sin ((of + <f>) (It 



^ai 



a^+(o 



2 [a • sin (wf + ^) — o> • cos ((of + <^ )] 



937. /^[e''' • sin ((of + <i>)'fdt 



■ sin \_(of + <^ — tan ^(w/a)]. 



4 



1 to • sin 2 ( o)/' 4- <^) + « ■ cos 2 (ojj' 



'^ + <^) ] 

a cf'^ + ft)"^ J 

1 cosr2a)?' + 2(^-tan-i(ai/Q:)] " 

(X 



Va"^ + 



U) 



'1 



114 



AUXILIARY FORMULAS. 



938. fie"' ■ COS (wt + <i»)ydt 



„2at 



1 o) • sin 2 (u)^ + ^) + (T • cos 2 (wt + ^) " 



a 



« 



«-^ + 



1 cos[2W + 2<^-taii-Xco/«)] "| 



V^7+^ 



In the case of a direct trigonometric function of (wt + <!>), 
T = 2 tt/o) is called the ^>erio(^ or the c//f^e. The mean 
value for any whole number of periods, reckoned from any 
epoch, of sin (wt + <^), cos (wt + (j>), or sin (cof + ^) • cos (wf + </>), 
is zero, whereas the mean value for any whole number of half 
periods, reckoned from any epoch, of either sin'-^ (wt + <f)) or 
cos^((ot + (f>) is one half. The mean value of sin (w.') from 
^ = to ;* = ^ T, or of cos (<ot) from - ^ T to + i T, is 2/7r 
or 0.6366. 

The mean value, for any number of whole periods, of either 
sin(a)?'+X) • sin(co^+/x) or cos(cu?'+X) • <-os(wf+fi) is h ■ cos(A— /x), 
while the mean value of sin(wi; + A) ■ cos ((Dt -\- /x) is ^ sin (\ — fi). 



TABLES. 115 



INTERPOLATION. 

If values of an analytic f unction, /(x), are given in a table for a number 
of values of the argument x, separated from one another consecutively by 
the constant small interval, 5, the differences between successive tabular 
values of the function are called ^rsi tabular differences, the differences of 
these first differences, second tabular differences, and so on. The tabular 
differences of the first, second, third, and fourth orders corresponding to 
z= a are 

Ai=/(a + 5)-/(a), 

A2 =/(a + 2 5) - 2 -/(a + 5) +/(a), 

A3 =f{a + 3 5) - 3 -/{a + 2 5) + 3 -/{a + 0) -f{a), 

A4 =/(a + 4 5) - 4 .f{a + 3 5) + 6 -/(a + 2 5) - 4 ./(a + 5) +/(a), 

where / (a) is any tabulated value. 

The value of the function for x = {a + h), v?here h = kS, is 

Z.X ^. V , * k{k-l) ^ k(k-l)(k-2) ^ 
f{a + h) =/(a) + A; • Ai + ^^^ • A2 + ^ ^ '- ■ A3 

_^ Mfc-l)(fc-2)(A:-3) _^_.^^^^ 



116 



TABLES. 



"he Probability Integral. 









1 


( 2 


'0 


dx. 
) 










X 





1 


2 


3 


4 


5 


6 


7 


8 


9 


0.00 


0.00000 


00113 


00226 


00339 


00451 


00564 


00677 


00790 


00903 


01016 


0.01 


0.01128 


01241 


01354 


01467 


01580 


01792 


01805 


01918 


02031 


02144 


0.02 


0.02256 


02369 


02482 


02595 


02708 


02820 


02933 


03046 


03159 


03271 


0.03 


0.03384 


03497 


03610 


03722 


03835 


03948 


04060 


04173 


04286 


04398 


0.04 


0.04511 


04624 


04736 


04849 


04962 


05074 


05187 


05299 


05412 


05525 


0.05 


0.05637 


05750 


05862 


05975 


06087 


06200 


06312 


06425 


06537 


06650 


0.06 


0.06762 


06875 


06987 


07099 


07212 


07324 


07437 


07549 


07661 


07773 


0.07 


0.07SS6 


07998 


08110 


08223 


08335 


08447 


08559 


08671 


08784 


08896 


0.08 


0.09008 


09120 


09232 


09344 


09456 


09568 


09680 


09792 


09904 


10016 


0.09 


0.10128 


10240 


10352 


10464 


10576 


10687 


10799 


10911 


11023 


11135 


0.10 


0.11246 


11358 


11470 


11581 


11693 


11S05 


11916 


12028 


12139 


12251 


0.11 


0.12362 


12474 


12585 


12697 


12808 


12919 


13031 


13142 


13253 


13365 


0.12 


0.13476 


135S7 


13698 


13809 


13921 


14032 


14143 


14254 


14365 


14476 


0.13 


0.14587 


14698 


14809 


14919 


15030 


15141 


15252 


15363 


15473 


15584 


0.14 


0.15695 


15805 


15916 


16027 


16137 


16248 


16358 


16468 


16579 


16689 


0.15 


0.16800 


16910 


17020 


17130 


17241 


17351 


17461 


17571 


17681 


17791 


0.16 


0.17901 


18011 


18121 


18231 


18341 


18451 


18560 


18670 


18780 


18890 


0.17 


0.18999 


19109 


19218 


19328 


19437 


19547 


19656 


19766 


19875 


19984 


0.18 


0.20094 


20203 


20312 


20421 


20530 


20639 


20748 


20857 


20966 


21075 


0.19 


0.21184 


21293 


21402 


21510 


21619 


21728 


21836 


21945 


22053 


22162 


0.20 


0.22270 


22379 


22487 


22595 


22704 


22812 


22920 


23028 


23136 


23244 


0.21 


0.23352 


23460 


23568 


23676 


23784 


23891 


23999 


24107 


24214 


24322 


0.22 


0.24430 


24537 


24645 


24752 


24859 


24967 


25074 


25181 


25288 


25395 


0.23 


0.25502 


25609 


25716 


25823 


25930 


26037 


26144 


26250 


26357 


26463 


0.24 


0.26570 


26677 


26783 


26889 


26996 


27102 


27208 


27314 


27421 


27527 


0.25 


0.27633 


27739 


27845 


27950 


28056 


28162 


28268 


28373 


28479 


28584 


0.26 


0.28690 


28795 


28901 


29006 


29111 


29217 


29322 


29427 


29532 


29637 


0.27 


0.29742 


29847 


29952 


30056 


30161 


30266 


30370 


30475 


30579 


30684 


0.28 


0.30788 


30892 


30997 


31101 


31205 


31309 


31413 


31517 


31621 


31725 


0.29 


0.31828 


31922 


32036 


32139 


32243 


32346 


32450 


32553 


32656 


32760 


0.30 


0.32863 


32966 


33069 


33172 


33275 


33378 


33480 


33583 


33686 


33788 


0.31 


0.33891 


33993 


34096 


34198 


34300 


34403 


34505 


34607 


34709 


34811 


0.32 


0.34913 


35014 


35116 


35218 


35319 


35421 


35523 


35624 


35725 


35827 


0.33 


0.35928 


36029 


36130 


36231 


36332 


36433 


36534 


36635 


36735 


36836 


0.34 


0.36936 


37037 


37137 


37238 


37338 


37438 


37538 


37638 


37738 


37838 


0.35 


0.37938 


38038 


38138 


38237 


38337 


38436 


38536 


38635 


38735 


38834 


0.36 


0.38933 


39032 


39131 


39230 


39329 


39428 


39526 


39625 


39724 


39822 


0.37 


0.39921 


40019 


40117 


40215 


40314 


40412 


40510 


40608 


40705 


40803 


0.38 


0.40901 


40999 


41096 


41194 


41291 


41388 


41486 


41583 


41680 


41777 


0.39 


0.41874 


41971 


42068 


42164 


42261 


42358 


42454 


42550 


42647 


42743 


0.40 


0.42839 


42935 


43031 


43127 


43223 


43319 


43415 


43510 


43606 


43701 


0.41 


0.43797 


43892 


43988 


44083 


44178 


44273 


44368 


44463 


44557 


44652 


0.42 


0.44747 


44841 


44936 


45030 


45124 


45219 


45313 


45407 


45501 


45595 


0.43 


0.45689 


45782 


45876 


45970 


46063 


46157 


46250 


46343 


46436 


46529 


0.44 


0.46623 


46715 


46808 


46901 


46994 


47086 


47179 


47271 


47364 


47456 


0.45 


0.47548 


47640 


47732 


47824 


47916 


48008 


48100 


48191 


48283 


48374 


0.46 


0.48466 


48557 


48648 


48739 


48830 


48921 


49012 


49103 


49193 


49284 


0.47 


0.49375 


49465 


49555 


49646 


49736 


49826 


49916 


50006 


50096 


50185 


0.48 


0.50275 


50365 


50454 


50543 


50633 


50722 


50811 


50900 


50989 


51078 


0.49 


0.51167 


51256 


51344 


51433 


51521 


51609 


51698 


51786 


51874 


51962 



1 



TABLES. 117 

The Probability Integral. 



123456789 

0.52050 52138 52226 52313 52401 52488 52576 52663 52750 52837 

0.52924 53011 53098 53185 53272 53358 53445 53531 53617 5370+ 

0.53790 53876 53962 54048 54134 54219 54305 54390 54476 54561 

0.54646 54732 54817 54902 54987 55071 55156 55241 55325 55410 

0.55494 55578 55662 55746 55830 55914 55998 56082 56165 56249 

0.56332 56416 56499 56582 56665 56748 56831 56914 56996 57079 

0.57162 57244 57326 57409 57491 57573 57655 57737 57818 57900 

0.57982 58063 58144 58226 58307 58388 58469 58550 58631 58712 

0.58792 58873 58953 59034 59114 59194 59274 59354 59434 59514 

0.59594 59673 59753 59832 59912 59991 60070 60149 60228 60307 

0.60386 60464 60543 60621 60700 60778 60856 60934 61012 61090 

0.61168 61246 61323 61401 61478 61556 61633 61710 61787 61864 

0.61941 62018 62095 62171 62248 62324 62400 62477 62553 62629 

0.62705 62780 62856 62932 63007 63083 63158 63233 63309 63384 

0.63459 63533 63608 63683 63757 63832 63906 63981 64055 64129 

0.64203 64277 64351 64424 64498 64572 64645 64718 64791 64865 

0.64938 65011 65083 65156 65229 65301 65374 65446 65519 65591 

0.65663 65735 65807 65878 65950 66022 66093 66165 66236 66307 

0.66378 66449 66520 66591 66662 66732 66803 66873 66944 67014 

0.67084 67154 67224 67294 67364 67433 67503 67572 67642 67711 

0.67780 67849 67918 67987 68056 68125 68193 68262 68330 68398 

0.68467 68535 68603 68671 68/38 68806 68874 68941 69009 69076 

0.69143 69210 69278 69344 69411 69478 69545 69611 6%78 69744 

0.69S10 69877 69943 70009 70075 70140 70206 70272 70337 70403 

0.70468 70533 70598 70663 70728 70793 70858 70922 70987 71051 

0.71116 71180 71244 71308 71372 71436 71500 71563 71627 71690 

0.71754 71817 71880 71943 72006 72069 72132 72195 72257 72320 

0.72382 72444 72507 72569 72631 72693 72755 72816 72878 72940 

0.73001 73062 73124 73185 73246 73307 73368 73429 73489 73550 

0.73610 73671 73731 73791 73851 73911 73971 74031 74091 74151 

0.74210 74270 74329 74388 74447 74506 74565 74624 74683 74742 

0.74800 74859 74917 74976 75034 75092 75150 75208 75266 75323 

0.75381 75439 75496 75553 75611 75668 75725 75782 75839 75896 

0.75952 76009 76066 76122 76178 76234 76291 76347 76403 76459 

0.76514 76570 76626 76681 76736 76792 76847 76902 76957 77012 

0.77067 77122 77176 77231 77285 77340 77394 77448 77502 77556 

0.77610 77664 77718 77771 77825 77878 77932 77985 78038 78091 

0.78144 78197 78250 78302 78355 78408 78460 78512 78565 78617 

0.78669 78721 78773 78824 78876 78928 78979 79031 79082 79133 

0.79184 79235 79286 79337 79388 79439 79489 79540 79590 79641 

0.79691 79741 79791 79841 79891 79941 79990 80040 80090 80139 

0.80188 80238 80287 80336 80385 80434 80482 80531 80580 80628 

0.80677 80725 80773 80822 80870 80918 80966 81013 81061 81109 

0.81156 81204 81251 81299 81346 81393 81440 81487 81534 81580 

0.81627 81674 81720 81767 81813 81859 81905 81951 81997 82043 

0.82089 82135 82180 82226 82271 82317 82362 82407 82452 82497 

0.82542 82587 82632 82677 82721 82766 82810 82855 82899 82943 

0.82987 83031 83075 83119 83162 83206 83250 83293 83337 83380 

0.83423 83466 83509 83552 83595 83638 83681 83723 83766 83808 

0.83851 83893 83935 83977 84020 84061 84103 84145 84187 84229 



118 



TABLES. 



The Probability Integral. 




"^ dx. 



X 





1 


2 


3 


4 


5 


6 


^ 

1 


8 


9 


1.00 


0.84270 


84312 


84353 


84394 


84435 


84477 


84518 


84559 


84600 


84640 


1.01 


0.84681 


84722 


84762 


84803 


84843 


84883 


84924 


84964 


85004 


85044 


1.02 


0.85084 


85124 


85163 


85203 


85243 


85282 


85322 


85361 


85400 


85439 


1.03 


0.85478 


85517 


85556 


85595 


85634 


85673 


85711 


85750 


85788 


85827 


1.04 


0.85865 


85903 


85941 


85979 


86017 


86055 


86093 


86131 


86169 


86206 


1.05 


0.86244 


86281 


86318 


86356 


86393 


86430 


86467 


86504 


86541 


86578 


1.06 


0.86614 


86651 


86688 


86724 


86760 


86797 


86833 


86869 


86905 


86941 


1.07 


0.86977 


87013 


87049 


87085 


87120 


87156 


87191 


87227 


87262 


87297 


1.08 


0.87333 


87368 


87403 


87438 


87473 


87507 


87542 


87577 


87611 


87646 


1.09 


0.87680 


87715 


87749 


87783 


87817 


87851 


87885 


87919 


87953 


87987 


1.10 


0.88021 


88054 


880SS 


88121 


88155 


SS188 


88221 


88254 


88287 


88320 


1.11 


0.88353 


88386 


88419 


88452 


88484 


88517 


88549 


88582 


88614 


88647 


1.12 


0.88679 


88711 


88743 


88775 


88807 


8SS39 


88871 


88902 


88934 


88966 


1.13 


0.88997 


89029 


89060 


89091 


89122 


89] 54 


89185 


89216 


89247 


89277 


1.14 


0.89308 


89339 


89370 


89400 


89431 


89461 


89492 


89522 


89552 


89582 


1.15 


0.89612 


89642 


89672 


89702 


89732 


89762 


89792 


89821 


89851 


89880 


1.16 


0.89910 


89939 


89968 


89997 


90027 


90056 


90085 


90114 


90142 


90171 


1.17 


0.90200 


90229 


90257 


90286 


90314 


90343 


90371 


90399 


90428 


90456 


1.18 


0.90484 


90512 


90540 


90568 


90595 


90623 


90651 


90678 


90706 


90733 


1.19 


0.90761 


90788 


90815 


90843 


90870 


90897 


90924 


90951 


90978 


91005 


1.20 


0.91031 


91058 


91085 


91111 


91138 


91164 


91191 


91217 


91243 


91269 


1.21 


0.91296 


91322 


91348 


91374 


91399 


91425 


91451 


91477 


91502 


91528 


1.22 


0.91553 


91579 


91604 


91630 


91655 


91680 


91705 


91730 


91755 


91780 


1.23 


0.91805 


91830 


91855 


91879 


91904 


91929 


91953 


91978 


92002 


92026 


1.24 


0.92051 


92075 


92099 


92123 


92147 


92171 


92195 


92219 


92243 


92266 


1.25 


0.92290 


92314 


92337 


92361 


92384 


92408 


92431 


92454 


92477 


92500 


1.26 


0.92524 


92547 


92570 


92593 


92615 


92638 


92661 


92684 


92706 


92729 


1.27 


0.92751 


92774 


92796 


92819 


92841 


92863 


92885 


92907 


92929 


92951 


1.28 


0.92973 


92995 


93017 


93039 


93061 


93082 


93104 


93126 


93147 


93168 


1.29 


0.93190 


93211 


93232 


93254 


93275 


93296 


93317 


93338 


93359 


93380 


1.30 


0.93401 


93422 


93442 


93463 


93484 


93504 


93525 


93545 


93566 


93586 


1.31 


0.93606 


93627 


93647 


93667 


93687 


93707 


93727 


93747 


93767 


93787 


1.32 


0.93807 


93826 


93846 


93866 


93885 


93905 


93924 


93944 


93963 


93982 


1.33 


0.94002 


94021 


94040 


94059 


94078 


94097 


94116 


94135 


94154 


94173 


1.34 


0.94191 


94210 


94229 


94247 


94266 


94284 


94303 


94321 


94340 


94358 


1.35 


0.94376 


94394 


94413 


94431 


94449 


94467 


94485 


94503 


94521 


94538 


1.36 


0.94556 


94574 


94592 


94609 


94627 


94644 


94662 


94679 


94697 


94714 


1.37 


0.94731 


94748 


94766 


94783 


94800 


94817 


94834 


94851 


94868 


94885 


1.38 


0.94902 


94918 


94935 


94952 


94968 


94985 


95002 


95018 


95035 


95051 


1.39 


0.95067 


95084 


95100 


95116 


95132 


95148 


95165 


95181 


95197 


95213 


1.40 


0.95229 


95244 


95260 


95276 


95292 


95307 


95323 


95339 


95354 


95370 


1.41 


0.95385 


95401 


95416 


95431 


95447 


95462 


95477 


95492 


95507 


95523 


1.42 


0.95538 


95553 


95568 


95582 


95597 


95612 


95627 


95642 


95656 


95671 


1.43 


0.95686 


95700 


95715 


95729 


95744 


95758 


95773 


95787 


95801 


95815 


1.44 


0.95830 


95844 


95858 


95872 


95886 


95900 


95914 


95928 


95942 


95956 


1.45 


0.95970 


95983 


95997 


96011 


96024 


96038 


96051 


96065 


96078 


96092 


1.46 


0.96105 


96119 


96132 


96145 


96159 


96172 


96185 


96198 


96211 


96224 


1.47 


0.96237 


96250 


96263 


96276 


96289 


96302 


96315 


96327 


96340 


96353 


1.48 


0.96365 


96378 


96391 


96403 


96416 


96428 


96440 


96453 


96465 


96478 


1.49 


0.96490 


96502 


96514 


96526 


96539 


96551 


96563 


96575 


96587 


96599 



TABLES. 



119 



The Probability Integral. 



X 


2 4 6 8 


X 


2 4 6 8 


1.50 


0.96611 96634 96658 96681 96705 


2.00 


0.99532 99536 99540 99544 99548 


1.51 


0.96728 96751 96774 96796 96819 


2.01 


0.99552 99556 99560 99564 99568 


1.52 


0.96841 96S64 96886 96908 96930 


2.02 


0.99572 99576 99580 99583 99587 


1.53 


0.96952 96973 96995 97016 97037 


2.03 


0.99591 99594 99598 99601 99605 


1.54 


0.97059 97080 97100 97121 97142 


2.04 


0.99609 99612 99616 99619 99622 


1.55 


0.97162 97183 97203 97223 97243 


2.05 


0.99626 99629 99633 99636 99639 


1.56 


0.97263 97283 97302 97322 97341 


2.06 


0.99642 99646 99649 99652 99655 


1.57 


0.97360 97379 9739S 97417 97436 


2.07 


0.99658 99661 99664 99667 99670 


1.58 


0.97455 97473 97492 97510 97528 


2. OS 


0.99673 99676 99679 996S2 99685 


1.59 


0.97546 97564 97582 97600 97617 


2.09 


0.99688 99691 99694 99697 99699 


1.60 


0.97635 97652 97670 97687 97704 


2.10 


0.99702 99705 99707 99710 99713 


1.61 


0.97721 97738 97754 97771 97787 


2.11 


0.99715 99718 99721 99723 99726 


1.62 


0.97804 97820 97836 97852 97868 


2.12 


0.99728 99731 99733 99736 99738 ' 


1.63 


0.978S4 97900 97916 97931 97947 


2.13 


0.99741 99743 99745 99748 99750 


1.64 


0.97962 97977 97993 98008 98023 


2.14 


0.99753 99755 99757 99759 99762 


1.65 


0.9S03S 98052 98067 98082 98096 


2.15 


0.99764 99766 99768 99770 99773 


1.66 


0.98110 98125 98139 98153 98167 


2.16 


0.99775 99777 99779 997S1 99783 


1.67 


0.98181 98195 98209 98222 98236 


2.17 


0.99785 99787 99789 99791 99793 


1.68 


0.98249 9S263 9S276 98289 98302 


2.18 


0.99795 99797 99799 99801 99803 


1.69 


0.9S315 98328 9S341 98354 98366 


2.19 


0.99805 99806 99808 99810 99812 


1.70 


0.98379 98392 98404 98416 98429 


2.20 


0.99814 99815 99817 99819 99821 


1.71 


0.98441 9S453 98465 98477 98489 


2.21 


0.99822 99824 99826 99827 99829 


1.72 


0.98500 98512 98524 98535 98546 


2.22 


0.99831 99832 99834 99836 99S37 


1.73 


0.98558 98569 98580 98591 98602 


2.23 


0.99839 99840 99842 99843 99845 


1.74 


0.98613 98624 98635 98646 98657 


2.24 


0.99846 99848 99849 99851 99852 


1.75 


0.98667 98678 98688 98699 98709 


2.25 


0.99854 99855 9985 7 99858 99859 


1.76 


0.98719 9S729 98739 98749 98759 


2.26 


0.99861 99862 99863 99865 99866 


1.77 


0.98769 98779 98789 98798 98808 


2.27 


0.99867 99869 99870 99871 99873 


1.78 


0.98817 98827 9SS36 98846 98855 


2.28 


0.99874 99875 99876 99877 99879 


1.79 


0.98864 98873 98882 98891 98900 


2.29 


0.99880 99881 99882 99883 99885 


1.80 


0.98909 98918 98927 98935 98944 


2.30 


0.99886 99887 99888 99889 99890 


1.81 


0.98952 98961 98969 98978 98986 


2.31 


0.99891 99892 99893 99894 99896 


1.82 


0.98994 99003 99011 99019 99027 


2.32 


0.99897 99898 99899 99900 99901 


1.83 


0.99035 99043 99050 99058 99066 


2.33 


0.99902 99903 99904 99905 99906 


1.84 


0.99074 99081 99089 99096 99104 


2.34 


0.99906 99907 99908 99909 99910 


1.85 


0.99111 99118 99126 99133 99140 


2.35 


0.99911 99912 99913 99914 99915 


1.86 


0.99147 99154 99161 99168 99175 


2.36 


0.99915 99916 99917 99918 99919 


1.87 


0.99182 99189 99196 99202 99209 


2.37 


0.99920 99920 99921 99922 99923 


l.SS 


0.99216 99222 99229 99235 99242 


2.38 


0.99924 99924 99925 99926 99927 


1.89 


0.99248 99254 99261 99267 99273 


2.39 


0.99928 99928 99929 99930 99930 


1.90 


0.99279 99285 99291 99297 99303 


2.40 


0.99931 99932 99933 99933 99934 


1.91 


0.99309 99315 99321 99326 99332 


2.41 


0.99935 99935 99936 99937 99937 


1.92 


0.99338 99343 99349 99355 99360 


2.42 


0.99938 99939 99939 99940 99940 


1.93 


0.99366 99371 99376 99382 99387 


2.43 


0.99941 99942 99942 99943 99943 


1.94 


0.99392 99397 99403 99408 99413 


2.44 


0.99944 99945 99945 99946 99946 


1.95 


0.99418 99423 99428 99433 99438 


2.45 


0.99947 99947 99948 99949 99949 


1.96 


0.99443 99447 99452 99457 99462 


2.46 


0.99950 99950 99951 99951 99952 


1.97 


0.99466 99471 99476 99480 99485 


2.47 


0.99952 99953 99953 99954 99954 


1.98 


0.99489 99494 99498 99502 99507 


2.48 


0.99955 99955 99956 99956 99957 


1.99 


0.99511 99515 99520 99524 99528 


2.49 


0.99957 99958 99958 99958 99959 


2.00 


0.99532 99536 99540 99544 99548 


2.50 


0.99959 99960 99960 99961 99961 



120 



TABLES. 



The Probability Integral. 

2 




-x2 






) 



X 





1 


2 


3 


4 


5 


6 


7 


8 


9 


2.5 


0.99959 


99961 


99963 


99965 


99967 


99969 


99971 


99972 


99974 


99975 


2.6 


0.99976 


99978 


99979 


99980 


99981 


99982 


99983 


99984 


99985 


99986 


2.7 


0.99987 


99987 


99988 


99989 


99989 


99990 


99991 


99991 


99992 


99992 


2.8 


0.99992 


99993 


99993 


99994 


99994 


99994 


99995 


99995 


99995 


99996 


2.9 


0.99996 


99996 


99996 


99997 


99997 


99997 


99997 


99997 


99997 


99998 


3.0 


0.99998 


99998 


99998 


99998 


99998 


99998 


99998 


99998 


99999 


99999 



The value, /, of the Probability Integral may always be found from the convergent series 

, 1 / afi a;5 a-' 
/= — = [x — , 



■v^V 3-l!^5-2! 7-3! 



■)■ 

but for large values of x, the semiconvergent series 

x-^-a\ 2x2 (2x2)3 (2x2)3^ ) 

is convenient. 



i 



TABLES. 



121 



Values of the Complete Elliptic Integrals, JTand E, for Different 
Values of the Modulus, k. 



K= p. ^^ ; E= pVl-A;2sin-z. 

»/o Vl — kr sin'' z *yo 



dz. 



sin-ifc 


E 


E 


sin-iA; 


K 


E 


1 
sin-ifc 


K 


E 


0° 


1.5708 


1.5708 


30° 


1.6858 


1.4675 


60° 


2.1565 


1.2111 


1° 


1.5709 


1.5707 


31° 


1.6941 


1.4608 


61° 


2.1842 


1.2015 


2° 


1.5713 


1.5703 


32° 


1.7028 


1.4539 


62° 


2.2132 


1.1920 


3° 


1.5719 


1.5697 


33° 


1.7119 


1.4469 


63° 


2.2435 


1.1826 


4° 


1.5727 


1.5689 


34° 


1.7214 


1.4397 


64° 


2.2754 


1.1732 


5° 


1.5738 


1.5678 


35° 


1.7312 


1,4323 


65° 


2.308S 


1.1638 


6° 


1.5751 


1.5665 


36° 


1.7415 


1.4248 


66° 


2.3439 


1.1545 


7° 


1.5767 


1.5649 


37° 


1.7522 


1.4171 


67° 


2.3809 


1.1453 


8° 


1.5785 


1.5632 


38° 


1.7633 


1.4092 


68° 


2.4198 


1.1362 


9° 


1.5S05 


1.5611 


39° 


1.7748 


1.4013 


69° 


2.4610 


1.1272 


10° 


1.5828 


1.5589 


40° 


1.7868 


1.3931 


70° 


2.5046 


1.1184 


11° 


1.5854 


1.5564 


41° 


1.7992 


1.3849 


71° 


2.5507 


1.1096 


12° 


1.5882 


1.5537 


42° 


1.8122 


1.3765 


72° 


2.5998 


1.1011 


13° 


1.5913 


1.5507 


43° 


1.8256 


1.3680 


73° 


2.6521 


1.0927 


14° 


1.5946 


1.5476 


44° 


1.8396 


1.3594 


74° 


2.7081 


1.0S44 


15° 


1.5981 


1.5442 


45° 


1.S541 


1.3506 


75° 


2.7681 


1.0764 


16° 


1.6020 


1.5405 


46° 


1.S691 


1.3418 


76° 


2.8327 


1.0686 


17° 


1.6061 


1.5367 


47° 


1.SS48 


1.3329 


77° 


2.9026 


1.0611 


18° 


1.6105 


1.5326 


48° 


1.9011 


1.3238 


78° 


2.9786 


1.0538 


19° 


1.6151 


1.5283 


49° 


1.9180 


1.3147 


79° 


3.0617 


1.0468 


20° 


1.6200 


1.5238 


50° 


1.9356 


1.3055 


80° 


3.1534 


1.0401 


21° 


1.6252 


1.5191 


51° 


1.9539 


1.2963 


81° 


3.2553 


1.0338 


22° 


1.6307 


1.5141 


52° 


1.9729 


1.2S70 


82° 


3.3699 


1.0278 


23° 


1.6365 


1.5090 


53° 


1.9927 


1.2776 


83° 


3.5004 


1.0223 


24° 


1.6426 


1.5037 


54° 


2.0133 


1.26S1 


84° 


3.6519 


1.0172 


25° 


1.6490 


1.4981 


55° 


2.0347 


1.2587 


85° 


3.8317 


1.0127 


26° 


1.6557 


1.4924 


56° 


2.0571 


1.2492 


86° 


4.0528 


1.0086 


27° 


1.6627 


1.4864 


57° 


2.080+ 


1.2397 


87° 


4.3387 


1.0053 


28° 


1.6701 


1.4803 


58° 


2.101-7 


1.2301 


88° 


4.7427 


1.0026 


29° 


1.6777 


1.4740 


59° 


2.1300 


1.2206 


89° 


5.4349 


1.0008 



122 



TABLES. 



Values of F{k, <t>) for Certain Values of k and ^t, 

dz 



^<*'*>=XV 



A;2 sin2 z 



<t> 


a = sin-^k. 


0° 


1(P 


15° 


30° 


45° 


60° 


75° 


80° 


90° 


1° 


0.0174 


0.0174 


0.0174 


0.0174 


0.0174 


0.0174 


0.0174 


0.0174 


0.0174 


2° 


0.0349 


0.0349 


0.0349 


0.0349 


0.0349 


0.0349 


0.0349 


0.0349 


0.0349 


3° 


0.0524 


0.0524 


0.0524 


0.0524 


0.0524 


0.0524 


0.0524 


0.0524 


0.0524 


40 


0.0698 


0.0698 


0.069S 


0.0698 


0.0698 


0.0699 


0.0699 


0.0699 


0.0699 


5° 


0.0873 


0.0873 


0.0873 


0.0873 


0.0873 


0.0S74 


0.0874 


0.0874 


0.0874 


10° 


0.1745 


0.1746 


0.1746 


0.1748 


0.1750 


0.1752 


0.1754 


0.1754 


0.1754 


15° 


0.2618 


0.2619 


0.2620 


0.2625 


0.2633 


0.2641 


0.2646 


0.2647 


0.2648 


20° 


0.3491 


0.3493 


0.3495 


0.3508 


0.3526 


0.3545 


0.3559 


0.3562 


0.3564 


25° 


0.4363 


0.4367 


0.4372 


0.4397 


0.4433 


0.4470 


0.4498 


0.4504 


0.4509 


30° 


0.5236 


0.5243 


0.5251 


0.5294 


0.5356 


0.5422 


0.5474 


0.5484 


0.5493 


35° 


0.6109 


0.6119 


0.6132 


0.6200 


0.6300 


0.6408 


0.6495 


0.6513 


0.6528 


40° 


0.6981 


0.6997 


0.7016 


0.7116 


0.7267 


0.7436 


0.7574 


0.7604 


0.7629 


45° 


0.7854 


0.7876 


0.7902 


0.8044 


0.8260 


0.8512 


0.8727 


0.8774 


0.8814 


50° 


0.8727 


0.8756 


0.8792 


0.8982 


0.9283 


0.9646 


0.9971 


1.0044 


1.0107 


55° 


0.9599 


0.9637 


0.9683 


0.9933 


1.0337 


1.0848 


1.1331 


1.1444 


1.1542 


60° 


l.(H72 


1.0519 


1.0577 


1.0896 


1.1424 


1.2125 


1.2837 


1.3014 


1.3170 


65° 


1.1345 


1.1402 


1.1474 


1.1869 


1.2545 


1.3489 


1.4532 


1.4810 


1.5064 


70° 


1.2217 


1.2286 


1.2373 


1.2853 


1.3697 


1.4944 


1.6468 


1.6918 


1.7354 


75° 


1.3090 


1.3171 


1.3273 


1.3846 


1.4S79 


1.6492 


1.8714 


1.9468 


2.0276 


80° 


1.3963 


1.4056 


1.4175 


1.4846 


1.6085 


1.8125 


2.1339 


2.2653 


2.4362 


85° 


1.4835 


1.4942 


1.5078 


1.5850 


1.7308 


1.9826 


2.4366 


2.6694 


3.1313 


86° 


1.5010 


1.5120 


1.5259 


1.6052 


1.7554 


2.0172 


2.5013 


2.7612 


3.3547 


87° 


1.5184 


1.5297 


1.5439 


1.6253 


1.7801 


2.0519 


2.5670 


2.8561 


3.6425 


88° 


1.5359 


1.5474 


1.5620 


1.6454 


1.8047 


2.0867 


2.6336 


2.9537 


4.0481 


89° 


1.5533 


1.5651 


1.5S01 


1.6656 


1.8294 


2.1216 


2.7007 


3.0530 


4.7414 


90° 


1.5708 


1.5828 


1.5981 


1.6858 


1.8541 


2.1565 


2.7681 


3.1534 


Inf. 



TABLES. 



123 



Values of E(k, 4>) for Certain Values of k and 0. 
E{k, ^) = I Vl - fc2 sin2 z • dz. 



: 


a = sin-i&. 


0° 


10° 


15° 


30° 


45° 


60° 


75° 


80° 


90° 


P 


0.0174 0.0174 


0.0174 


0.0174 


0.0174 


0.0174 


0.0174 


0.0174 


0.0174 


2» 


0.0349 0.0349 


0.0349 


0.0349 


0.0349 


0.0349 


0.0349 


0.0349 


0.0349 


3» 


0.0524 0.0524 


0.0524 


0.0524 


0.0524 


0.0523 


0.0523 


0.0523 


0.0523 


40 


0.0698 


0.0698 


0.0698 


0.0698 


0.0698 


0.0698 


0.0698 


0.0698 


0.0698 


5° 


0.0873 


0.0873 


0.0873 


0.0872 


0.0872 


0.0872 


0.0S72 


0.0872 


0.0872 


10° 


0.1745 


0.1745 


0.1745 


0.1743 


0.1741 


0.1739 


0.1737 


0.1737 


0.1736 


15° 


0.2618 


0.2617 


0.2616 


0.2611 


0.2603 


0.2596 


0.2590 


0.2589 


0.2588 


20° 


0.3491 


0.3489 


0.3486 


0.3473 


0.3456 


0.3438 


0.3425 


0.3422 


0.3420 


25° 


0.4363 


0.4359 


0.4354 


0.4330 


0.4296 


0.4261 


0.4236 


0.4230 


0.4226 


30° 


0.5236 


0.5229 


0.5221 


0.5179 


0.5120 


0.5061 


0.5016 


0.5007 


0.5000 


35° 


0.6109 


0.6098 


0.6085 


0.6019 


0.5928 


0.5833 


0.5762 


0.5748 


0.5736 


40° 


0.6981 


0.6966 


0.6947 


0.6851 


0.6715 


0.6575 


0.6468 


0.6446 


0.6428 


45° 


0.7854 


0.7832 


0.7806 


0.7672 


0.7482 


0.7282 


0.7129 


0.7097 


0.7071 


50° 


0.8727 


0.8698 


0.8663 


0.8483 


0.8226 


0.7954 


0.7741 


0.7697 


0.7660 


55° 


0.9599 


0.9562 


0.9517 


0.9284 


0.8949 


0.8588 


0.8302 


0.8242 


0.8192 


60° 


1.0472 


1.0426 


1.0368 


1.0076 


0.9650 


0.9184 


0.8808 


0.8728 


0.8660 


65° 


1.1345 


1.12SS 


1.1218 


1.0858 


1.0329 


0.9743 


0.9258 


0.9152 


0.9063 


70° 


1.2217 


1.2149 


1.2065 


1.1632 


1.0990 


1.0266 


0.9652 


0.9514 


0.9397 


75° 


1.3090 


1.3010 


1.2911 


1.2399 


1.1635 


1.0759 


0.9992 


0.9814 


0.9659 


80° 


1.3963 


1.3S70 


1.3755 


1.3161 


1.2266 


1.1225 


1.0282 


1.0054 


0.9848 


85° 


1.4835 


1.4729 


1.4598 


1.3919 


1.2889 


1.1673 


1.0534 


1.0244 


0.9962 


86° 


1.5010 


1.4901 


1.4767 


1.4070 


1.3012 


1.1761 


1.0581 


1.0277 


0.9976 


87° 


1.5184 


1.5073 


1.4936 


1.4221 


1.3136 


1.1848 


1.0628 


1.0309 


0.9986 


88° 


1.5359 


1.5245 


1.5104 


1.4372 


1.3260 


1.1936 


1.0674 


1.0340 


0.9994 


89° 


1.5533 


1.5417 


1.5273 


1.4524 


1.3383 


1.2023 


1.0719 


1.0371 


0.9998 


90« 


1.5708 


1.5589 


1.5442 


1.4675 


1.3506 


1.2111 


1.0764 


1.0401 

1 


1.0000 



124 



TABLES. 



Hyperbolic Functions. 



1, 


e^ 


e-^ 


sinhx 


coshx 


gdx 


0.00 


1.0000 


1.0000 


0.0000 


1.0000 


o!oooo 


.01 


1.0100 


0.9900 


.0100 


1.0000 


0.5729 


.02 


1.0202 


.9802 


.0200 


1.0002 


1.1458 


.03 


1.0305 


.9704 


.0300 


1.0004 


1.7186 


.04 


1.0408 


.9608 


.0400 


1.0008 


2.2912 


.05 


1.0513 


.9512 


.0500 


1.0013 


2.8636 


.06 


1.0618 


.9418 


.0600 


1.0018 


3.4357 


.07 


1.0725 


.9324 


.0701 


1.0025 


4.0074 


.08 


1.0833 


.9231 


.0801 


1.0032 


4.5788 


.09 


1.0942 


.9139 


.0901 


1.0041 


5.1497 


.10 


1.1052 


.9048 


.1002 


1.0050 


5.720 


.11 


1.1163 


.8958 


.1102 


1.0061 


6.290 


.12 


1.1275 


.8869 


.1203 


1.0072 


6.859 


.13 


1.1388 


.8781 


.1304 


1.0085 


7.428 


.14 


1.1503 


.8694 


.1405 


1.0098 


7.995 


.15 


1.1618 


.8607 


.1506 


1.0113 


8.562 


.16 


1.1735 


.8521 


.1607 


1.0128 


9.128 


.17 


1.1853 


.8437 


.1708 


1.0145 


9.694 


.18 


1.1972 


.8353 


.1810 


1.0162 


10.258 


.19 


1.2092 


.8270 


.1911 


1.0181 


10.821 


.20 


1.2214 


.8187 


.2013 


1.0201 


11.384 


.21 


1.2337 


.8106 


.2115 


1.0221 


11.945 


.22 


1.2461 


.8025 


.2218 


1.0243 


12.505 


.23 


1.2586 


.7945 


.2320 


1.0266 


13.063 


.24 


1.2712 


.7866 


.2423 


1.0289 


13.621 


.25 


1.2840 


.7788 


.2526 


1.0314 


14.177 


.26 


1.2969 


.7711 


.2629 


1.0340 


14.732 


.27 


1.3100 


.7634 


.2733 


1.0367 


15.285 


.28 


1.3231 


.7558 


.2837 


1.0395 


15.837 


.29 


1.3364 


.7483 


.2941 


1.0423 


16.388 


.30 


1.3499 


.7408 


.3045 


1.0453 


16.937 


.31 


1.3634 


.7334 


.3150 


1.0484 


17.484 


.32 


1.3771 


.7261 


.3255 


1.0516 


18.030 


.33 


1.3910 


.7189 


.3360 


1.0549 


18.573 


.34 


1.4049 


.7118 


.3466 


1.0584 


19.116 


.35 


1.4191 


.7047 


.3572 


1.0619 


19.656 


.36 


1.4333 


.6977 


.3678 


1.0655 


20.195 


.37 


1.4477 


.6907 


.3785 


1.0692 


20.732 


.38 


1.4623 


.6839 


.3892 


1.0731 


21.267 


.39 


1.4770 


.6771 


.4000 


1.0770 


21.800 


.40 


1.4918 


.6703 


.4108 


1.0811 


22.331 


.41 


1.5068 


.6636 


.4216 


1.0852 


22.859 


.42 


1.5220 


.6570 


.4325 


1.0895 


23.386 


.43 


1.5373 


.6505 


.4434 


1.0939 


23.911 


.44 


1.5527 


.6440 


.4543 


1.0984 


24.434 


.45 


1.5683 


.6376 


.4653 


1.1030 


24.955 


.46 


1.5841 


.6313 


.4764 


1.1077 


25.473 


.47 


1.6000 


.6250 


.4875 


1.1125 


25.989 


.48 


1.6161 


.6188 


.4986 


1.1174 


26.503 


.49 


1.6323 


.6126 


.5098 


1.1225 


27.015 


0.50 


1.648f 


0.6065 


0.5211 


1.1276 


27?524 



Note. —This table is talien from Prof. Byerly's Treatise on Fourier's Series, published by Messrs. 
Oiim& Co. 



TABLES. 



125 



Hyperbolic Functions. 



X 


e-^ 


er-x 


sinhx 


coshx 


gdx 


0.50 


1.6487 


0.6065 


0.5211 


1.1276 


27!524 


.51 


1.6653 


.6005 


.5324 


1.1329 


28.031 


.52 


1.6820 


.5945 


.5438 


1.1383 


28.535 


.53 


1.6989 


.5886 


.5552 


1.1438 


29.037 


.54 


1.7160 


.5827 


.5666 


1.1494 


29.537 


.55 


1.7333 


.5770 


.5782 


1.1551 


30.034 


.56 


1.7507 


.5712 


.5897 


1.1609 


30.529 


.57 


1.7683 


.5655 


.6014 


1.1669 


31.021 


.58 


1.7860 


.5599 


.6131 


1.1730 


31.511 


.59 


1.8040 


.5543 


.6248 


1.1792 


31.998 


.60 


1.8221 


.5488 


6367 


1.1855 


32.483 


.61 


1.8404 


.5433 


.6485 


1.1919 


32.965 


.62 


1.8589 


.5379 


.6605 


1.1984 


33.444 


.63 


1.S776 


.5326 


.6725 


1.2051 


33.921 


.64 


1.8965 


.5273 


.6846 


1.2119 


34.395 


.65 


1.9155 


.5220 


.6967 


1.2188 


34.867 


.66 


1.9348 


.5169 


.7090 


1.2258 


35.336 


.67 


1.9542 


.5117 


.7213 


1.2330 


35.802 


-68 


1.9739 


.5066 


.7336 


1.2402 


36.265 


-69 


1.9937 


.5016 


.7461 


1.2476 


36.726 


.70 


2.0138 


.4966 


.7586 


1.2552 


37.183 


-71 


2.0340 


.4916 


.7712 


1.2628 


37.638 


-72 


2.0544 


.4867 


.7838 


1.2706 


38.091 


-73 


2.0751 


.4819 


.7966 


1.2785 


38.540 


.74 


2.0959 


.4771 


.8094 


1.2865 


38.987 


.75 


2.1170 


.4724 


.8223 


1.2947 


39.431 


.76 


2.1383 


.4677 


.8353 


1.3030 


39.872 


-77 


2.1598 


.4630 


.8484 


1.3114 


40.310 


.78 


2.1815 


.4584 


.8615 


1.3199 


40.746 


-79 


2.2034 


.4538 


.8748 


1.3286 


41.179 


-80 


2.2255 


.4493 


.8881 


1.3374 


41.608 


-81 


2.2479 


.4449 


.9015 


1.3464 


42.035 


.82 


2.2705 


.4404 


.9150 


1.3555 


42.460 


-83 


2.2933 


.4360 


.9286 


1.3647 


42.881 


-84 


2.3164 


.4317 


.9423 


1.3740 


43.299 


-85 


2.3396 


.4274 


.9561 


1.3835 


43.715 


.86 


2.3632 


.4232 


.9700 


1.3932 


44.128 


.87 


2.3869 


.4190 


.9840 


1.4029 


44.537 


.88 


2.4109 


.4148 


.9981 


1.4128 


44.944 


.89 


2.4351 


.4107 


1.0122 


1.4229 


45.348 


.90 


2.4596 


.4066 


1.0265 


1.4331 


45.750 


.91 


2.4843 


.4025 


1.0409 


1.4434 


46.148 


-92 


2.5093 


.3985 


1.0554 


1.4539 


46.544 


-93 


2.5345 


.3946 


1.0700 


1.4645 


46.936 


-94 


2.5600 


.3906 


1.0847 


1.4753 


47.326 


-95 


2.5857 


.3867 


1.0995 


1.4862 


47.713 


-96 


2.6117 


.3829 


1.1144 


1.4973 


48.097 


.97 


2.6379 


.3791 


1.1294 


1.5085 


48.478 


.98 


2.6645 


.3753 


1.1446 


1.5199 


48.857 


.99 


2.6912 


.3716 


1.1598 


1.5314 


49.232 


1.00 


2.7183 


0.3679 


1.1752 


1.5431 


49!60S 



siiih X = tan gd x ; cosh a: = sec gd a; ; tanh x = sin gd x. 



126 



TABLES. 



Hyperbolic Functions. 



X 


I .si nil X 


I cosh X 


X 


isinhx 


I cosh X 


X 


isinhx 


I cosh X 


1.00 


0.0701 


0.1884 


1.50 


0.3282 


0.3715 


2.00 


0.5595 


0.5754 


1.01 


.0758 


.1917 


1.51 


.3330 


.3754 


2.01 


.5640 


.5796 


1.02 


.0815 


.1950 


1.52 


.3378 


.3794 


2.02 


.5685 


.5838 


1.03 


.0871 


.1984 


1.53 


.3426 


.3833 


2.03 


.5730 


.5880 


1.04 


.0927 


.2018 


1.54 


.3474 


.3873 


2.04 


.5775 


.5922 


1.05 


.0982 


.2051 


1.55 


.3521 


.3913 


2.05 


.5820 


.5964 


1.06 


.1038 


.2086 


1.56 


.3569 


.3952 


2.06 


.5865 


.6006 


1.07 


.1093 


.2120 


1.57 


.3616 


.3992 


2.07 


.5910 


.6048 


1.08 


.1148 


.2154 


1.58 


.3663 


.4032 


2.08 


.5955 


.6090 


1.09 


.1203 


.2189 


1.59 


.3711 


.4072 


2.09 


.6000 


.6132 


1.10 


.1257 


.2223 


1.60 


.3758 


.4112 


2.10 


.6044 


.6175 


1.11 


.1311 


.2258 


1.61 


.3805 


.4152 


2.11 


.6089 


.6217 


1.12 


.1365 


.2293 


1.62 


.3852 


.4192 


2.12 


.6134 


.6259 


1.13 


.1419 


.2328 


1.63 


.3899 


.4232 


2.13 


.6178 


.6301 


1.14 


.1472 


.2364 


1.64 


.3946 


.4273 


2.14 


.6223 


.6343 


1.15 


.1525 


.2399 


1.65 


.3992 


.4313 


2.15 


.6268 


.6386 


1.16 


.1578 


.2435 


1.66 


.4039 


.4353 


2.16 


.6312 


.6428 


1.17 


.1631 


.2470 


1.67 


.4086 


.4394 


2.17 


.6357 


.6470 


1.18 


.1684 


.2506 


1.68 


.4132 


.4434 


2.18 


.6401 


.6512 


1.19 


.1736 


.2542 


1.69 


.4179 


.4475 


2.19 


.6446 


.6555 


1.20 


.1788 


.2578 


1.70 


.4225 


.4515 


2.20 


.6491 


.6597 


1.21 


.1840 


.2615 


1.71 


.4272 


.4556 


2.21 


.6535 


.6640 


1.22 


.1892 


.2651 


1.72 


.4318 


.4597 


2.22 


.6580 


.6682 


1.23 


.1944 


.2688 


1.73 


.4364 


.4637 


2.23 


.6624 


.6724 


1.24 


.1995 


.2724 


1.74 


.4411 


.4678 


2.24 


.6668 


.6767 


1.25 


.2046 


.2761 


1.75 


.4457 


.4719 


2.25 


.6713 


.6809 


1.26 


.2098 


.2798 


1.76 


.4503 


.4760 


2.26 


.6757 


.6852 


1.27 


.2148 


.2835 


1.77 


.4549 


.4801 


2.27 


.6802 


.6894 


1.28 


.2199 


.2872 


1.78 


.4595 


.4842 


2.28 


.6846 


.6937 


1.29 


.2250 


.2909 


1.79 


.4641 


.4883 


2.29 


.6890 


.6979 


1.30 


.2300 


.2947 


1.80 


.4687 


.4924 


2.30 


.6935 


.7022 


1.31 


.2351 


.2984 


LSI 


.4733 


.4965 


2.31 


.6979 


.7064 


1.32 


.2401 


.3022 


1.82 


.4778 


.5006 


2.32 


.7023 


.7107 


1.33 


.2451 


.3059 


1.83 


.4824 


.5048 


2.33 


.7067 


.7150 


1.34 


.2501 


.3097 


1.84 


.4870 


.5089 


2.34 


.7112 


.7192 


1.35 


.2551 


.3135 


1.85 


.4915 


.5130 


2.35 


.7156 


.7235 


1.36 


.2600 


.3173 


1.86 


.4961 


.5172 


2.36 


.7200 


.7278 


1.37 


.2650 


.3211 


1.87 


.5007 


.5213 


2.37 


.7244 


.7320 


1.38 


.2699 


.3249 


1.88 


.5052 


.5254 


2.38 


.7289 


.7363 


1.39 


.2748 


.3288 


1.89 


.5098 


.5296 


2.38 


.7333 


.7406 


1.40 


.2797 


.3326 


1.90 


.5143 


.5337 


2.40 


.7377 


.7448 


1.41 


.2846 


.3365 


1.91 


.5188 


.5379 


2.41 


.7421 


.7491 


1.42 


.2895 


.3403 


1.92 


.5234 


.5421 


2.42 


.7465 


.7534 


1.43 


.2944 


.3442 


1.93 


.5279 


.5462 


2.43 


.7509 


.7577 


1.44 


.2993 


-3481 


1.94 


.5324 


.5504 


2.44 


.7553 


.7619 


1.45 


.3041 


.3520 


1.95 


.5370 


.5545 


2.45 


.7597 


.7662 


1.46 


.3090 


.3559 


1.96 


.5415 


.5687 


2.46 


.7642 


.7705 


1.47 


.3138 


.3598 


1.97 


.5460 


.5629 


2.47 


.7686 


.7748 


1.48 


.3186 


.3637 


1.98 


.5505 


.5671 


2.48 


.7730 


.7791 


1.49 


.3234 


.3676 


1.99 


.5550 


.5713 


2.49 


.7774 


.7833 


1.50 


0.3282 


0.3715 


2.00 


0.5595 


0.5754 


2.50 


0.7818 


0.7876 



TABLES. 



127 



Hyperbolic Functions. 



X 


I sinh X 


I cosh X 


X 


isinhx 


I cosh X 


X 


I sinh X 


I cosh X 


2.50 


0.7S1S 


0.7876 


2.1 S 


0.8915 


0.8951 


3.0 


1.0008 


1.0029 


2.51 


.7862 


.7919 


2.76 


.8959 


.8994 


3.1 


1.0444 


1.0462 


2.52 


.7906 


.7962 


2.77 


.9003 


.9037 


3.2 


1.0880 


1.0894 


2.53 


.7950 


.8005 


2.78 


.9046 


.9080 


3.3 


1.1316 


1.1327 


2.54 


.7994 


.8048 


2.79 


.9090 


.9123 


3.4 


1.1751 


1.1761 


2.55 


.8038 


.8091 


2.80 


.9134 


.9166 


3.5 


1.2186 


1.2194 


2.56 


.8082 


.8134 


2.81 


.9178 


.9209 


3.6 


1.2621 


1.2628 


2.57 


.8126 


.8176 


2.82 


.9221 


.9252 


3.7 


1.3056 


1.3061 


2.58 


.8169 


.8219 


2.83 


.9265 


.9295 


3.8 


1.3491 


1.3495 


2.59 


.8213 


.8262 


2.84 


.9309 


.9338 


3.9 


1.3925 


1.3929 


2.60 


.8257 


.8305 


2.85 


.9353 


.9382 


4.0 


1.4360 


1.4363 


2.61 


.8301 


.8348 


2.86 


.9396 


.9425 


4.1 


1.4795 


1.4797 


2.62 


.8345 


.8391 


2.87 


.9440 


.9468 


4.2 


1.5229 


1.5231 


2.63 


.8389 


.8434 


2.88 


.9484 


.9511 


4.3 


1.5664 


1.5665 


2.64 


.8433 


.8477 


2.89 


.9527 


.9554 


4.4 


1.6098 


1.6099 


2.65 


.8477 


.8520 


2 90 


.9571 


.9597 


4.5 


1.6532 


1.6533 


2.66 


.8521 


.8563 


2.91 


.9615 


.9641 


4.6 


1.6967 


1.6968 


2.67 


.8564 


.8606 


2.92 


.9658 


.9684 


4.7 


1.740] 


1.7402 


2.68 


.8608 


.8649 


2.93 


.9702 


.9727 


4.8 


1.7836 


1.7836 


2.69 


.8652 


.8692 


2.94 


.9746 


.9770 


4.9 


1.8270 


1.8270 


2.70 


.8696 


.8735 


2.95 


.9789 


.9813 


5.0 


1.8704 


1.8705 


2.71 


.8740 


.8778 


2.96 


.9833 


.9856 


6.0 


2.3047 


2.3047 


2.72 


.8784 


.8821 


2.97 


.9877 


.9900 


7.0 


2.7390 


2.7390 


2.73 


.8827 


.8864 


2.98 


.9920 


.9943 


8.0 


3.1733 


3.1733 


2.74 


.8871 


.8907 


2.99 


.9964 


.9986 


9.0 


3.6076 


3.6076 


2.75 


0.8915 


0.8951 


3.00 


1.0008 


1.0029 


10.0 


4.0419 


4.0419 



For values of x greater than 7.0, we may write, to five places of deci- 
mals at least, 

logio sinh X = logio cosh x = log i e^ = x (0.4342945) + 1.6989700. 



The Values of e-x^ for Certain Values of x. 



X 


e-^ 


X 


e-^ 


X 


Q-X 


X 


Q-X 


1/10 


0.90484 


8/10 


0.44933 


18/10 


0.16530 


5 


0.00674 


1/8 


0.88250 


9/10 


0.40657 


2 


0.13534 


11/2 


0.00409 


1/6 


0.84648 


1 


0.36788 


9/4 


0.10540 


6 


0.00248 


2/10 


0.81873 


11/10 


0.33287 


5/2 


0.08209 


13/2 


0.00150 


1/4 


0.77880 


9/8 


0.32465 


8/3 


0.06948 


7 


0.00091 


3/10 


0.74082 


12/10 


0.30119 


3 


0.04979 


15/2 


0.00055 


1/3 


0.71653 


5/4 


0.28650 


25/8 


0.04394 


8 


0.00034 


4/10 


0.67032 


13/10 


0.27253 


16/5 


0.04076 


9 


0.00012 


5/10 


0.60653 


4/3 


0.26360 


18/5 


0.02732 


10 


0.00004 


6/10 


0.54881 


14/10 


0.24660 


4 


0.01832 


11 


0.00002 


2/3 


0.51342 


3/2 


0.22313 


25/6 


0.01550 


12 


0.00001 


7/10 


0.49659 


16/10 


0.20190 


9/2 


0.01111 


13 


0.00000 



128 



TABLES. 



The Common Logarithms of e^ and e-«. 




« 


logioe^ 


logioe-^ 




0.00001 


0.0000043429 


1.9999956571 




0.00002 


0.0000086859 


1.9999913141 




0.00003 


0.0000130288 


1.9999869712 




0.00004 


0.0000173718 


1.9999826282 




0.00005 


0.0000217147 


1.9999782853 




0.00006 


0.0000260577 


1.9999739423 




0.00007 


0.0000304006 


1.9999695994 • 




0.00008 


0.0000347436 


1.9999652564 




0.00009 


0.0000390865 


1.9999609135 




0.00010 


0.0000434294 


1.9999565706 




0.00020 


0.0000868589 


1.9999131411 




0.00030 


0.0001302883 


1.9998697117 




0.00040 


0.0001737178 


1.9998262822 




0.00050 


0.0002171472 


1.9997828528 




0.00060 


0.0002605767 


1.9997394233 




0.00070 


0.0003040061 


1.9996959939 




0.00080 


0.0003474356 


1.9996525644 




0.00090 


0.0003908650 


1.9996091350 




0.00100 


0.0004342945 


1.9995657055 




0.00200 


0.0008685890 


1.9991314110 




0.00300 


0.0013028834 


1.9986971166 




0.00400 


0.0017371779 


1.9982628221 




0.00500 


0.0021714724 


1.9978285276 




0.00600 


0.0026057669 


1.9973942331 




0.00700 


0.0030400614 


1.9969599386 




0.00800 


0.0034743559 


1.9965256441 




0.00900 


0.0039086503 


1.9960913497 




0.01000 


0.0043429448 


1.9956570552 




0.02000 


0.0086858896 


1.9913141 ICH- 




0.03000 


0.0130288345 


1.9869711655 




0.04000 


0.0173717793 


T.9826282207 




0.05000 


0.0217147241 


1.9782852759 




0.06000 


0.0260576689 


1.9739423311 




0.07000 


0.0304006137 


1.9695993863 





TABLES. 



129 





X 


logio e* 


logio e-" 






0.08000 


0.0347435586 


1.9652564414 






0.09000 


0.0390865034 


1.9609134966 






0.10000 


0.0434294482 


1.9565705518 






0.20000 


0.0868588964 


1.9131411036 






0.30000 


0.1302883446 


1.8697116554 






0.40000 


0.1737177928 


1.8262822072 






0.50000 


02171472410 


1.7828527590 






0.60000 


0.2605766891 


1.7394233109 






0.70000 


0.3040061373 


1.6959938627 






0.80000 


0.3474355855 


1.6525644145 






090000 


0.3908650337 


1.6091349663 






1.00000 


0.4342944819 


1.5657055181 






2.00000 


0.8685889638 


1.1314110362 






3.00000 


1.3028834457 


2.6971165543 






4.00000 


1.7371779276 


2.2628220724 






5.00000 


2.1714724095 


3.8285275905 






6.00000 


2.6057668914 


3.3942331086 






7.00000 


3.0400613733 


4.9599386267 






8.00000 


3.4743558552 


4.5256441448 






9.00000 


3.9086503371 


4.0913496629 






10.00000 


4.3429448190 


5.6570551810 






20.00000 


8.6858896381 


9.3141103619 






30.00000 


13.0288344571 


14.9711655429 ^ 






40.00000 


17.3717792761 


18.6282207239 






50.00000 


21.7147240952 


22.2852759048 






60.00000 


26.0576689142 


27.9423310858 






70.00000 


30.4006137332 


31.5993862668 






80.00000 


34.7435585523 


35.2564414477 






90.00000 


39.0865033713 


40.9134966287 






100.00000 


43.4294481903 


44.5705518097 






200.00000 


86.8588963807 


87.1411036193 






300.00000 


130.2883445710 


131.7116554290 






400.00000 


173.7177927613 


174.2822072387 






500.00000 


217.1472409516 


218.8527590tS4 





Note •- log e^ + V = log e* + log &>. Thus, log giis-iirs - 49.139465 ISa 



130 



TABLES. 



Five-Place Natural Logarithms. 



No. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 


1.00 


0.0 0000 


0100 


0200 


0300 


0399 


0499 


0598 


0698 


0797 


0896 


100-99 


1.01 


0.0 0995 


1094 


1193 


1292 


1390 


1489 


1587 


1686 


1784 


1882 


99-98 


1.02 


0.0 1980 


2078 


2176 


2274 


2372 


2469 


2567 


2664 


2762 


2859 


98-97 


1.03 


0.0 2956 


3053 


3150 


3247 


3343 


3440 


3537 


3633 


3730 


3826 


97-96 


1.04 


0.0 3922 


4018 


4114 


4210 


4306 


4402 


4497 


4593 


4688 


4784 


96-95 


1.05 


0.0 4879 


4974 


5069 


5164 


5259 


5354 


5449 


5543 


5638 


5733 


95-94 


1.06 


0.0 5827 


5921 


6015 


6110 


6204 


6297 


6391 


6485 


6579 


6672 


94 


1.07 


0.0 6766 


6859 


6953 


7046 


7139 


7232 


7325 


7418 


7511 


7603 


93 


1.08 


0.0 7696 


7789 


7881 


7973 


8066 


8158 


8250 


8342 


8434 


8526 


93-92 


1.09 


0.0 8618 


8709 


8801 


8893 


8984 


9075 


9167 


9258 


9349 


9430 


92-91 


1.10 


0.0 9531 


9622 


9713 


9803 


9894 


9985 


*0075 


0165 


0256 


0346 


91-90 


1.11 


0.1 0436 


0526 


0616 


0706 


0796 


0885 


0975 


1065 


1154 


1244 


90-89 


1.12 


0.1 1333 


1422 


1511 


1600 


1689 


1778 


1867 


1956 


2045 


2133 


89 


1.13 


0.1 2222 


2310 


2399 


2487 


2575 


2663 


2751 


2839 


2927 


3015 


88 


1.14 


0.1 3103 


3191 


3278 


3366 


3453 


3540 


3628 


3715 


3802 


3889 


88-87 


1.15 


0.1 3976 


4063 


4150 


4237 


4323 


4410 


4497 


4583 


4669 


4756 


87-86 


1.16 


0.1 4842 


4928 


5014 


5100 


5186 


5272 


5358 


5444 


5529 


5615 


86 


1.17 


0.1 5700 


5786 


5871 


5956 


6042 


6127 


6212 


6297 


6382 


6467 


85 


1.18 


0.16551 


6636 


6721 


6805 


6890 


6974 


7059 


7143 


7227 


7311 


85-84 


1.19 


0.1 7395 


7479 


7563 


7647 


7731 


7815 


7898 


7982 


8065 


8149 


84-83 


1.20 


0.1 8232 


8315 


8399 


8482 


8565 


8648 


8731 


8814 


8897 


9979 


83 


1.21 


0.1 9062 


9145 


9227 


9310 


9392 


9474 


9557 


9639 


9721 


9803 


83-82 


1.22 


0.1 9885 


9967 *0049 


0131 


0212 


0294 


0376 


0457 


0539 


0620 


82-81 


1.23 


0.2 0701 


0783 


0864 


0945 


1026 


1107 


1188 


1269 


1350 


1430 


81 


1.24 


0.21511 


1592 


1672 


1753 


1833 


1914 


1994 


2074 


2154 


2234 


81-80 


1.25 


0.2 2314 


2394 


2474 


2554 


2634 


2714 


2793 


2873 


2952 


3032 


80-79 


1.26 


0.2 3111 


3191 


3270 


3349 


3428 


3507 


3586 


3665 


3744 


3823 


79 


1.27 


0.2 3902 


3980 


4059 


4138 


4216 


4295 


4373 


4451 


4530 


4608 


79-78 


1.28 


0.2 4686 


4764 


4842 


4920 


4998 


5076 


5154 


5231 


5309 


5387 


78 


1.29 


0.2 5464 


5542 


5619 


5697 


5774 


5811 


5928 


6005 


6082 


6159 


77 


1.30 


0.2 6236 


6313 


6390 


6467 


6544 


6620 


6697 


6773 


6850 


6926 


77-76 


1.31 


0.2 7003 


7079 


7155 


7231 


7308 


7384 


7460 


7536 


7612 


7687 


76 


1.32 


0.2 7763 


7839 


7915 


7990 


8066 


8141 


8217 


8292 


8367 


8443 


76-75 


1.33 


0.2 8518 


8593 


8668 


8743 


8818 


8893 


8968 


9043 


9118 


9192 


75 


1.34 


0.2 9267 


9342 


9416 


9491 


9565 


9639 


9714 


9788 


9862 


9936 


75-74 


1.35 


0.3 0010 


0085 


0158 


0232 


0306 


0380 


0454 


0528 


0601 


0675 


74 


1.36 


0.3 0748 


0822 


0895 


0969 


1042 


1115 


1189 


1262 


1335 


1408 


74-73 


1.37 


0.3 1481 


1554 


1627 


1700 


1773 


1845 


1918 


1991 


2063 


2136 


73-72 


1.38 


0.3 2208 


2281 


2353 


2426 


2498 


2570 


2642 


2714 


2786 


2858 


72 


1.39 


0.3 2930 


3002 


3074 


3146 


3218 


3289 


3361 


3433 


3504 


3576 


72-71 


1.40 


0.3 3647 


3719 


3790 


3861 


3933 


4004 


4075 


4146 


4217 


4288 


71 


1.41 


0.3 4359 


4430 


4501 


4572 


4642 


4713 


4784 


4854 


4925 


4995 


71-70 


1.42 


0.3 5066 


5136 


5206 


5277 


5347 


5417 


5487 


5557 


5677 


5697 


70 


1.43 


0.3 5767 


5837 


5907 


5977 


6047 


6116 


6186 


6256 


6335 


6395 


70-69 


1.44 


0.3 6464 


6534 


6603 


6672 


6742 


6811 


6880 


6949 


7018 


7087 


69 


1.45 


0.3 7156 


7225 


7294 


7363 


7432 


7501 


7569 


7638 


7707 


7775 


69 


1.46 


0.3 7844 


7912 


7981 


8049 


8117 


8186 


8254 


8322 


8390 


8458 


68 


1.47 


0.3 8526 


8594 


8662 


8730 


8798 


8866 


8934 


9001 


9069 


9137 


68 


1.48 


0.3 9204 


9272 


9339 


9407 


9474 


9541 


9609 


9676 


9743 


9810 


68-67 


1.49 


0.3 9878 


9945 


*0012 


0079 


0146 


0213 


0279 


0346 


0413 


0480 


67 


1.50 


0.4 0547 


0613 


0680 


0746 


0813 


0879 


0946 


1012 


1078 


1145 


67-€6 







1 


2 


3 


4 


5 


6 


7 


8 


9 





TABLES. 



131 



Five-Place Natural Logarithms. 



No. 





1 


2 


3 


4 


5 





7 


8 


9 


D. 


1.50 


0.4 0547 


0613 


0680 


0746 


0813 


0879 


0946 


1012 


1078 


1145 


67-66 


1.51 


0.41211 


1277 


1343 


1409 


1476 


1542 


1608 


1673 


1739 


1805 


66 


1.52 


0.4 1871 


1937 


2003 


2068 


2134 


2199 


2265 


2331 


2396 


2461 


66-65 


1.53 


0.4 2527 


2592 


2657 


2723 


2788 


2853 


2918 


2983 


3048 


3113 


65 


1.54 


0.4 3178 


3243 


3308 


3373 


3438 


3502 


3567 


3632 


3696 


3761 


65-64 


1.55 


0.4 3825 


3890 


3954 


4019 


4083 


4148 


4212 


4276 


4340 


4404 


64 


1.56 


0.4 4469 


4533 


4597 


4661 


4725 


4789 


4852 


4916 


4980 


5044 


64 


1.57 


0.4 5108 


5171 


5235 


5298 


5362 


5426 


5489 


5552 


5616 


5679 


64-63 


1.58 


0.4 5742 


5S06 


5869 


5932 


5995 


6058 


6122 


6185 


6248 


6310 


63 


1.59 


0.4 6373 


6436 


6499 


6562 


6625 


6687 


6750 


6813 


6875 


6938 


63 


1.60 


0.4 7000 


7063 


7125 


7188 


7250 


7312 


7375 


7437 


7499 


7561 


62 


1.61 


0.4 7623 


7686 


7748 


7810 


7872 


7933 


7995 


8057 


8119 


8181 


63 


1.62 


0.4 8243 


8304 


8366 


8428 


8489 


8551 


8612 


8674 


8735 


8797 


62-61 


1.63 


0.4 8858 


8919 


8981 


9042 


9103 


9164 


9225 


9287 


9348 


9409 


61 


1.64 


0.4 9470 


9531 


9592 


9652 


9713 


9774 


9835 


9896 


9956 *0017 


61 


1.65 


0.5 0078 


013S 


0199 


0259 


0320 


0380 


0441 


0501 


0561 


0622 


61-60 


1.66 


■ 0.5 0682 


0742 


0802 


0862 


0922 


0983 


1043 


1103 


1163 


1222 


60 


1.67 


0.5 1282 


1342 


1402 


1462 


1522 


1581 


1641 


1701 


1760 


1820 


60 


1.68 


0.5 1879 


1939 


1998 


2058 


2117 


2177 


2236 


2295 


2354 


2414 


60-59 


1.69 


0.5 2473 


2532 


2591 


2650 


2709 


2768 


2827 


2886 


2945 


3004 


59 


1.70 


0.5 3063 


3122 


3180 


3239 


3298 


3357 


3415 


3474 


3532 


3591 


59-58 


1.71 


0.5 3649 


3708 


3766 


3825 


3883 


3941 


4000 


4058 


4116 


4174 


58 


1.72 


0.5 4232 


4291 


4349 


4407 


4465 


4523 


4581 


4639 


4696 


4754 


58 


1.73 


0.5 4812 


4870 


4928 


4985 


5043 


5101 


5158 


5216 


5274 


5331 


58-57 


1.74 


0.5 5389 


5446 


5503 


5561 


5618 


5675 


5/oj 


5790 


5847 


5904 


57 


1.75 


0.5 5962 


6019 


6076 


6133 


6190 


6247 


6304 


6361 


6418 


6475 


57 


1.76 


0.5 6531 


6588 


6645 


6702 


6758 


6815 


6872 


6928 


6985 


7041 


57 


1.77 


0.5 7098 


7154 


7211 


7267 


7324 


7380 


7436 


7493 


7549 


7605 


56 


1.78 


0.5 7661 


7718 


7774 


7830 


7886 


7942 


7998 


8054 


8110 


8166 


56 


1.79 


0.5 8222 


8277 


8333 


8389 


8445 


8501 


8556 


8612 


8667 


8723 


56 


1.80 


0.5 8779 


8834 


8890 


8945 


9001 


9056 


9111 


9167 


9222 


9277 


56-55 


1.81 


0.5 9333 


9388 


9443 


9498 


9553 


9609 


9664 


9719 


9774 


9829 


55 


1.82 


0.5 9884 


9939 


9993 


*004S 


0103 


0158 


0213 


0268 


0322 


0377 


55 


1.83 


0.6 0432 


0486 


0541 


0595 


0650 


0704 


0759 


0813 


0868 


0922 


55-54 


1.84 


0.6 0977 


1031 


1085 


1139 


1194 


1248 


1302 


1356 


1410 


1464 


54 


1.85 


0.61519 


1573 


1627 


1681 


1735 


1788 


1842 


1896 


1950 


2004 


54 


1.86 


0.6 2058 


2111 


2165 


2219 


2272 


2326 


2380 


2433 


2487 


2540 


54-53 


1.87 


0.6 2594 


2647 


2701 


2754 


2808 


2861 


2914 


2967 


3021 


3074 


53 


1.88 


0.6 3127 


3180 


3234 


3287 


3340 


3393 


3446 


3499 


3552 


3605 


53 


1.89 


0.6 3658 


3711 


3763 


3816 


3869 


3922 


3975 


4027 


4080 


4133 


53 


1.90 


0.6 4185 


4238 


4291 


4343 


4396 


4448 


4501 


4553 


4606 


4658 


53-52 


1.91 


0.6 4710 


4763 


4815 


4867 


4920 


4972 


5024 


5076 


5128 


5180 


52 


1.92 


0.6 5233 


5285 


5337 


5389 


5441 


5493 


5545 


5596 


5648 


5700 


52 


1.93 


0.6 5752 


5804 


5856 


5907 


5959 


6011 


6062 


6114 


6166 


6217 


52 


1.94 


0.6 6269 


6320 


6372 


6423 


6475 


6526 


6578 


6629 


6680 


6732 


52-51 


1.95 


0.6 6783 


6834 


6885 


6937 


6988 


7039 


7090 


7141 


7192 


7243 


51 


1.96 


0.6 7294 


7345 


7396 


7447 


7498 


7549 


7600 


7651 


7702 


7753 


51 


1.97 


0.6 7803 


7854 


7905 


7956 


8006 


8057 


8107 


8158 


8209 


8259 


51 


1.98 


0.6 8310 


8360 


8411 


8461 


8512 


8562 


8612 


8663 


8713 


8763 


50 


1.99 


0.6 8813 


8864 


8914 


8964 


9014 


9064 


9115 


9165 


9215 


9265 


50 


2.00 


0.6 9315 


9365 


9415 


9465 


9515 


9564 


9614 


9664 


9714 


9764 


50 







1 


2 


3 


4 


5 


6 


7 


8 


9 





132 



TABLES. 
Five-Place Natural Logarithms. 



No. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 


2.00 


0.6 9315 


9365 


9415 


9465 


9515 


9564 


9614 


9664 


9714 


9764 


50 


2.01 


0.6 9813 


9863 


9913 


9963 - 


*^0012 


0062 


0112 


0161 


0211 


0260 


50 


2.02 


0.7 0310 


0359 


0409 


0458 


0508 


0557 


0606 


0656 


0705 


0754 


49 


2.03 


0.7 0804 


0853 


0902 


0951 


1000 


1050 


1099 


1148 


1197 


1246 


49 


2.04 


0.7 1295 


1344 


1393 


1442 


1491 


1540 


1589 


1638 


1686 


1735 


49 


2.05 


0.7 1784 


1833 


1881 


1930 


1979 


2028 


2076 


2125 


2173 


2222 


49 


2.06 


0.7 2271 


2319 


2368 


2416 


2465 


2513 


2561 


2610 


2658 


2707 


49-48 


2.07 


0.7 2755 


2803 


2851 


2900 


2948 


2996 


3044 


3092 


3141 


3189 


48 


2.08 


0.7 3237 


3285 


3333 


3381 


3429 


3477 


3525 


3573 


3621 


3669 


48 


2.09 


0.7 3716 


3764 


3812 


3860 


3908 


3955 


4003 


4051 


4098 


4146 


48 


2.10 


0.7 4194 


4241 


4289 


4336 


4384 


4432 


4479 


4527 


4574 


4621 


48-47 


2.11 


0.7 4669 


4716 


4764 


4811 


4858 


4905 


4953 


5000 


5047 


5094 


47 


2.12 


0.7 5142 


5189 


5236 


5283 


5330 


5377 


5424 


5471 


5518 


5565 


47 


2.13 


0.7 5612 


5659 


5706 


5753 


5800 


5847 


5893 


5940 


5987 


6034 


47 


2.14 


0.7 6081 


6127 


6174 


6221 


6267 


6314 


6361 


6407 


6454 


6500 


47 


2.LS 


0.7 6547 


6593 


6640 


6686 


6733 


6779 


6825 


6872 


6918 


6965 


47-46 


2.16 


0.7 7011 


7057 


7103 


7150 


7196 


7242 


7288 


7334 


7381 


7427 


46 


2.17 


0.7 7473 


7519 


7565 


7611 


7657 


7703 


7749 


7795 


7841 


7887 


46 


2.18 


0.7 7932 


7978 


8024 


8070 


8116 


8162 


8207 


8253 


8299 


8344 


46 


2.19 


0.7 8390 


8436 


8481 


8527 


8573 


8618 


8664 


8709 


8755 


8800 


46-45 


2.20 


0.7 8846 


8891 


8937 


8982 


9027 


9073 


9118 


9163 


9209 


9254 


45 


2.21 


0.7 9299 


9344 


9390 


9435 


9480 


9525 


9570 


9615 


9661 


9706 


45 


2.22 


0.7 9751 


9796 


9841 


9886 


9931 


9976 *0021 


0066 


0110 


0155 


45 


2.23 


0.8 0200 


0245 


0290 


0335 


0379 


0424 


0469 


0514 


0558 


0603 


45 


2.24 


0.8 0648 


0692 


0737 


0781 


0826 


0871 


0915 


0960 


1004 


1049 


45-44 


2.25 


0.8 1093 


1137 


1182 


1226 


1271 


1315 


1359 


1404 


1448 


1492 


44 


2.26 


0.8 1536 


1581 


1625 


1669 


1713 


1757 


1802 


1846 


1890 


1934 


44 


2.27 


0.8 1978 


2022 


2066 


2110 


2154 


2198 


2242 


2286 


2330 


2374 


44 


2.28 


0.8 2418 


2461 


2505 


2549 


2593 


2637 


2680 


2724 


2768 


2812 


44 


2.29 


0.8 2855 


2899 


2942 


2986 


3030 


3073 


3117 


3160 


3204 


3247 


44-43 


2.30 


0.8 3291 


3334 


3378 


3421 


3465 


3508 


3551 


3595 


3638 


3681 


43 


2.31 


0.8 3725 


3768 


3811 


3855 


3898 


3941 


3984 


4027 


4070 


4114 


43 


2.32 


0.8 4157 


4200 


4243 


4286 


4329 


4372 


4415 


4458 


4501 


4544 


43 


2.33 


0.8 4587 


4630 


4673 


4715 


4758 


4801 


4844 


4887 


4930 


4972 


43 


2.34 


0.8 5015 


5058 


5101 


5143 


5186 


5229 


5271 


5314 


5356 


5399 


43 


2.35 


0.8 5442 


5484 


5527 


5569 


5612 


5654 


5697 


5739 


5781 


5824 


43-48 


2.36 


0.8 5866 


5909 


5951 


5993 


6036 


6078 


6120 


6162 


6205 


6247 


42 


2.37 


0.8 6289 


6331 


6373 


6415 


6458 


6500 


6542 


6584 


6626 


6668 


42 


2.38 


0.8 6710 


6752 


6794 


6836 


6878 


6920 


6962 


7004 


7046 


7087 


42 


2.39 


0.8 7129 


7171 


7213 


7255 


7297 


7338 


7380 


7422 


7464 


7505 


42 


2.40 


0.8 7547 


7589 


7630 


7672 


7713 


7755 


7797 


7838 


7880 


7921 


42 


2.41 


0.8 7963 


8004 


8046 


8087 


8129 


8170 


8211 


8253 


8294 


8335 


41 


2.42 


0.8 8377 


8418 


8459 


8501 


8542 


8583 


8624 


8666 


8707 


8748 


41 


2.43 


0.8 8789 


8830 


8871 


8913 


8954 


8995 


9036 


9077 


9118 


9159 


41 


2.44 


0.8 9200 


9241 


9282 


9323 


9364 


9405 


9445 


9486 


9527 


9568 


41 


2.45 


0.8 9609 


9650 


9690 


9731 


9772 


9813 


9853 


9894 


9935 


9975 


41 


2.46 


0.9 0016 


0057 


0097 


0138 


0179 


0219 


0260 


0300 


0341 


0381 


41-40 


2.47 


0.9 0422 


0462 


0503 


0543 


0584 


0624 


0664 


0705 


0745 


0786 


40 


2.48 


0.9 0826 


0866 


0906 


0947 


0987 


1027 


1067 


1108 


1148 


1188 


40 


2.49 


0.9 1228 


1268 


1309 


1349 


1389 


1429 


1469 


1509 


1549 


1589 


40 


2.60 


0.91629 


1669 


1709 


1749 


1789 


1829 


1869 


1909 


1949 


1988 


40 






J 


1 


2 


3 


4 


5 


6 


7 


8 


9 





I 



TABLES. 



133 



Five-Place Natural Logarithms. 



No. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 


2.50 


0.91629 


1669 


1709 


1749 


1789 


1829 


1869 


1909 


1949 


1988 


40 


2.51 


0.9 2028 


2068 


2108 


2148 


2188 


2227 


2267 


2307 


2346 


2386 


40 


2.52 


0.9 2426 


2466 


2505 


2545 


2584 


2624 


2664 


2703 


2743 


2782 


40 


2.53 


0.9 2S22 


2S61 


2901 


2940 


2980 


3019 


3059 


3098 


3138 


3177 


40-39 


2.54 


0.9 3216 


3256 


3295 


3334 


3374 


3413 


3452 


3492 


3531 


3570 


39 


2.55 


0.9 3609 


3649 


3688 


3727 


3766 


3805 


3844 


3883 


3923 


3962 


39 


2.56 


0.9 4001 


4040 


4079 


4118 


4157 


4196 


4235 


4274 


4313 


4352 


39 


2.57 


0.9 4391 


4429 


4468 


4507 


4546 


4585 


4624 


4663 


4701 


4740 


39 


2.58 


0.9 4779 


4818 


4856 


4895 


4934 


4973 


5011 


5050 


5089 


5127 


39 


2.59 


0.9 5166 


5204 


5243 


5282 


5320 


5359 


5397 


5436 


5474 


5513 


39-38 


2.60 


0.9 5551 


5590 


5628 


5666 


5705 


5743 


5782 


5820 


5858 


5897 


38 


2.61 


0.9 5935 


5973 


6012 


6050 


6088 


6126 


6165 


6203 


6241 


6279 


38 


2.62 


0.9 6317 


6356 


6394 


6432 


6470 


6508 


6546 


6584 


6622 


6660 


38 


2.63 


0.9 6698 


6736 


6774 


6812 


6S50 


6S88 


6926 


6964 


7002 


7040 


38 


2.64 


0.9 7078 


7116 


7154 


7191 


7229 


7267 


7305 


7343 


7380 


7418 


38 


2.65 


0.9 7456 


7494 


7531 


7569 


7607 


7644 


7682 


7720 


7757 


7795 


38 


2.66 


0.9 7833 


7S70 


7908 


7945 


7983 


8020 


8058 


8095 


8133 


8170 


38-37 


2.67 


0.9 8208 


8245 


8283 


8320 


8358 


8395 


8432 


8470 


8507 


8544 


37 


2.68 


0.9 8582 


8619 


8656 


8694 


8731 


8768 


8805 


8843 


8880 


8917 


37 


2.69 


0.9 8954 


8991 


9028 


9066 


9103 


9140 


9177 


9214 


9251 


9288 


37 


2.70 


0.9 9325 


9362 


9399 


9436 


9473 


9510 


9547 


9584 


9621 


9658 


37 


2.71 


0.9 9695 


9732 


9769 


9806 


9842 


9879 


9916 


9953 


9990 


*0026 


37 


2.72 


1.0 0063 


0100 


0137 


0173 


0210 


0247 


0284 


0320 


0357 


0394 


37 


2.73 


1.0 0430 


0467 


0503 


0540 


0577 


0613 


0650 


0686 


0723 


0759 


37 


2.74 


1.0 0796 


0832 


0869 


0905 


0942 


0978 


1015 


1051 


1087 


1124 


36 


2.75 


1.0 1160 


1196 


1233 


1269 


1305 


1342 


1378 


1414 


1451 


1487 


36 


2.76 


1.0 1523 


1559 


1596 


1632 


1668 


1704 


1740 


1776 


1813 


1849 


36 


2.77 


1.0 1S85 


1921 


1957 


1993 


2029 


2065 


2101 


2137 


2173 


2209 


36 


2.78 


1.0 2245 


2281 


2317 


2353 


2389 


2425 


2461 


2497 


2532 


2588 


36 


2.79 


1.0 2604 


2640 


2676 


2712 


2747 


2783 


2819 


2855 


2890 


2926 


36 


2.80 


1.0 2962 


2998 


3033 


3069 


3105 


3140 


3176 


3212 


3247 


3283 


36 


2.81 


1.0 3318 


3354 


3390 


3425 


3461 


3496 


3532 


3567 


3603 


3638 


36-35 


2.82 


1.0 3674 


3709 


3745 


3780 


3815 


3851 


3886 


3922 


3957 


3992 


35 


2.S3 


1.0 4028 


4063 


4098 


4134 


4169 


4204 


4239 


4275 


4310 


4345 


35 


2.84 


1.0 43 SO 


4416 


4451 


4486 


4521 


4556 


4591 


4627 


4662 


4697 


35 


2.85 


1.0 4732 


4767 


4802 


4837 


4872 


4907 


4942 


4977 


5012 


5047 


35 


2.86 


1.0 5082 


5117 


5152 


5187 


5222 


5257 


5292 


5327 


5361 


5396 


35 


2.87 


1.0 5431 


5466 


5501 


5536 


5570 


5605 


5640 


5675 


5710 


5744 


35 


2.88 


1.0 5779 


5814 


5848 


5883 


5918 


5952 


5987 


6022 


6056 


6091 


35 


2.89 


1.0 6126 


6160 


6195 


6229 


6264 


6299 


6333 


6368 


6402 


6437 


35-34 


2.90 


1.0 6471 


6506 


6540 


6574 


6609 


6643 


6678 


6712 


6747 


6781 


34 


2.91 


1.0 6815 


6850 


6884 


6918 


6953 


6987 


7021 


7056 


7090 


7124 


34 


2.92 


1.0 7158 


7193 


7227 


7261 


7295 


7329 


7364 


7398 


7432 


7466 


34 


2.93 


1.0 7500 


7534 


7568 


7603 


7637 


7671 


7705 


7739 


7773 


7807 


34 


2.94 


1.0 7841 


7875 


7909 


7943 


7977 


8011 


8045 


8079 


8113 


8147 


34 


2.95 


1.0 8181 


8214 


8248 


8282 


8316 


8350 


8384 


8418 


8451 


8485 


34 


2.96 


1.0 8519 


8553 


8586 


8620 


8654 


8688 


8721 


8755 


8789 


8823 


34 


2.97 


1.0 8856 


8890 


8924 


8957 


8991 


9024 


9058 


9092 


9125 


9159 


34 


2.98 


1.0 9192 


9226 


9259 


9293 


9326 


9360 


9393 


9427 


9460 


9494 


34-33 


2.99 


1.0 9527 


9561 


9594 


9628 


9661 


9694 


9728 


9761 


9795 


9828 


33 


3.00 


1.0 9861 


9895 


9928 


9961 


9994 


*0028 


0061 


0094 


0128 


0161 


33 







1 


2 


3 


4 


5 


6 


7 


8 


9 





134 



TABLES. 



Five-Place Natural Logarithms. 



No, 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 


3.00 


1.0 9861 


9895 


9928 


9961 


9994 


*0028 


0061 


0094 


0128 


0161 


33 


3.01 


1 10194 


0227 


0260 


0294 


0327 


0360 


0393 


0426 


0459 


0493 


33 


3.02 


i.l 0526 


0559 


0592 


0625 


0658 


0691 


0724 


0757 


0790 


0823 


33 


3.03 


1.1 0856 


0889 


0922 


0955 


0988 


1021 


1054 


1087 


1120 


1153 


33 


3.04 


1.1 1186 


1219 


1252 


1284 


1317 


1350 


1383 


1416 


1449 


1481 


33 


3.05 


1.11514 


1547 


1580 


1612 


1645 


1678 


1711 


1743 


1776 


1809 


33 


3.06 


1.1 1841 


1874 


1907 


1939 


1972 


2005 


2037 


2070 


2103 


2135 


33 


3.07 


1.1 2168 


2200 


2233 


2265 


2298 


2330 


2363 


2396 


2428 


2460 


33-33 


3.08 


1.1 2493 


2525 


2558 


2590 


2623 


2655 


2688 


2720 


2752 


2785 


32 


3.09 


1.12817 


2849 


2882 


2914 


2946 


2979 


3011 


3043 


3076 


3108 


32 


8.10 


1.13140 


3172 


3205 


3237 


3269 


3301 


3334 


3366 


3398 


3430 


32 


3.11 


1.1 3462 


3494 


3527 


3559 


3591 


3623 


3655 


3687 


3719 


3751 


32 


3.12 


1.1 3783 


3815 


3847 


3879 


3911 


3943 


3955 


4007 


4039 


4071 


32 


3.13 


1.1 4103 


4135 


4167 


4199 


4231 


4263 


4295 


4327 


4359 


4390 


32 


3.14 


1.14422 


4454 


4486 


4518 


4550 


4581 


4613 


4645 


4677 


4708 


32 


3.15 


1.1 4740 


4772 


4804 


4835 


4867 


4899 


4931 


4962 


4994 


5026 


32 


3.16 


1.1 5057 


5089 


5120 


5152 


5184 


5215 


5247 


5278 


5310 


5342 


32 


3.17 


1.15373 


5405 


5436 


5468 


5499 


5531 


5562 


5594 


5625 


5657 


32-31 


3.18 


1.1 5688 


5720 


5751 


5782 


5814 


5845 


5877 


5908 


5939 


5971 


31 


3.19 


1.1 6002 


6033 


6065 


6096 


6127 


6159 


6190 


6221 


6253 


6284 


31 


3.20 


1.1 6315 


6346 


6378 


6409 


6440 


6471 


6502 


6534 


6565 


6596 


31 


3.21 


1.1 6627 


6658 


6689 


6721 


6752 


6783 


6814 


6845 


6876 


6907 


31 


3.22 


1.1 6938 


6969 


7000 


7031 


7062 


7093 


7124 


7155 


7186 


7217 


31 


3.23 


1.1 7248 


7279 


7310 


7341 


7372 


7403 


7434 


7465 


7496 


7526 


31 


3.24 


1.1 7557 


7588 


7619 


7650 


7681 


7712 


7742 


7773 


7804 


7835 


31 


3.25 


1.1 7865 


7896 


7927 


7958 


7989 


8019 


8050 


8081 


8111 


8142 


31 


3.26 


1.18173 


8203 


8234 


8265 


8295 


8326 


8357 


8387 


8418 


8448 


31 


3.27 


1.1 8479 


8510 


8540 


8571 


8601 


8632 


8662 


8693 


8723 


8754 


31-30 


3.28 


1.1 8784 


8815 


8845 


8876 


8906 


8937 


8967 


8998 


9028 


9058 


30 


3.29 


1.1 9089 


9119 


9150 


9180 


9210 


9241 


9271 


9301 


9332 


9362 


30 


3.30 


1.1 9392 


9423 


9453 


9483 


9513 


9544 


9574 


9604 


9634 


9665 


30 


3.31 


1.1 9695 


9725 


9755 


9785 


9816 


9846 


9876 


9906 


9936 


9966 


30 


3.32 


1.1 9996 


*0027 


0057 


0087 


0117 


0147 


0177 


0207 


0237 


0267 


30 


3.33 


1.2 0297 


0327 


0357 


0387 


0417 


0447 


0477 


0507 


0537 


0567 


30 


3.34 


1.2 0597 


0627 


0657 


06S7 


0717 


0747 


0777 


0806 


0836 


0866 


30 


3.35 


1.2 0896 


0926 


0956 


0986 


1015 


1045 


1075 


1105 


1135 


1164 


30 


3.36 


1.21194 


1224 


1254 


1283 


1313 


1343 


1373 


1402 


1432 


1462 


30 


3.37 


1.2 1491 


1521 


1551 


1580 


1610 


1640 


1669 


1699 


1728 


1758 


30 


3.38 


1.2 1788 


1817 


1847 


1876 


1906 


1935 


1965 


1994 


2024 


2053 


30 


3.39 


1.2 2083 


2112 


2142 


2171 


2201 


2230 


2260 


2289 


2319 


2348 


29 


3.40 


1.2 2378 


2407 


2436 


2466 


2495 


2524 


2554 


2583 


2613 


2642 


29 


3.41 


1.2 2671 


2701 


2730 


2759 


2788 


2818 


2847 


2876 


2906 


2935 


29 


3.42 


1.2 2964 


2993 


3023 


3052 


3081 


3110 


3139 


3169 


3198 


3227 


29 


3.43 


1.2 3256 


3285 


3314 


3343 


3373 


3402 


3431 


3460 


3489 


3518 


29 


3.44 


1.2 3547 


3576 


3605 


3634 


3663 


3692 


3721 


3750 


3779 


3808 


29 


3.45 


1.2 3837 


3866 


3895 


3924 


3953 


3982 


4011 


4040 


4069 


4098 


29 


3.46 


1.2 4127 


4156 


4185 


4214 


4242 


4271 


4300 


4329 


4358 


4387 


29 


3.47 


1.2 4415 


4444 


4473 


4502 


4531 


4559 


4588 


4617 


4646 


4674 


29 


3.48 


1.2 4703 


4732 


4761 


4789 


4818 


4847 


4875 


4904 


4933 


4962 


29 


3.49 


1.2 4990 


5019 


5047 


5076 


5105 


5133 


5162 


5191 


5219 


5248 


29 


8.50 


1.2 5276 


5305 


5333 


5362 


5391 


5419 


5448 


5476 


5505 


5533 


29-28 







1 


2 


3 


4 


5 


6 


7 


8 


9 





TABLES. 
Five-Place Natural Logarithms. 



135 



No. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 


3.50 


1.2 5276 


5305 


5333 


5362 


5391 


5419 


5448 


5476 


5505 


5533 


29-28 


3.51 


1.2 5562 


5590 


5619 


5647 


5675 


5704 


5732 


5761 


5789 


5818 


28 


3.52 


1.2 5846 


5875 


5903 


5931 


5960 


5988 


6016 


6045 


6073 


6101 


28 


3.53 


1.2 6130 


6158 


6186 


6215 


6243 


6271 


6300 


6328 


6356 


6384 


28 


3.54 


1.2 6413 


6441 


6469 


6497 


6526 


6554 


6582 


6610 


6638 


6667 


28 


3.55 


1.2 6695 


6723 


6751 


6779 


6807 


6836 


6864 


6892 


6920 


6948 


28 


3.56 


1.2 6976 


7004 


7032 


7060 


7088 


7116 


7144 


7172 


7201 


7229 


28 


3.57 


1.2 7257 


7285 


7313 


7341 


7369 


7397 


7424 


7452 


7480 


7508 


28 


3.5S 


1.2 7536 


7564 


7592 


7620 


7648 


7676 


7704 


7732 


7759 


7787 


28 


3.59 


1.2 7815 


7843 


7871 


7899 


7927 


7954 


7982 


8010 


8038 


8066 


28 


3.60 


1.2 8093 


8121 


8149 


8177 


8204 


8232 


8260 


8288 


8315 


8343 


28 


3.61 


1.2 8371 


8398 


8426 


8454 


8482 


8509 


8537 


8564 


8592 


8620 


28 


3.62 


1.2 8647 


8675 


8703 


8730 


8758 


8785 


8813 


8841 


8868 


8896 


28 


3.63 


1.2 8923 


8951 


8978 


9006 


9033 


9061 


9088 


9116 


9143 


9171 


28-27 


3.64 


1.2 9198 


9226 


9253 


9281 


9308 


9336 


9363 


9390 


9418 


9445 


27 


3.65 


1.2 9473 


9500 


9527 


9555 


9582 


9610 


9637 


9664 


9692 


9719 


27 


3.66 


1.2 9746 


9774 


9801 


9828 


9856 


9883 


9910 


9937 


9965 


9992 


27 


3.67 


1.3 0019 


0046 


0074 


0101 


0128 


0155 


0183 


0210 


0237 


0264 


27 


3.68 


1.3 0291 


0318 


0346 


0373 


0400 


0427 


0454 


0481 


0508 


0536 


27 


3.69 


1.3 0563 


0590 


0617 


0644 


0671 


0698 


0725 


0752 


0779 


0806 


27 


3.70 


1.3 0833 


0860 


0887 


0914 


0941 


0968 


0995 


1022 


1049 


1076 


27 


3.71 


1.3 1103 


1130 


1157 


1184 


1211 


1238 


1265 


1292 


1319 


1345 


27 


3.72 


1.3 1372 


1399 


1426 


1453 


1480 


1507 


1534 


1560 


1587 


1614 


27 


3.73 


1.3 1641 


1668 


1694 


1721 


1748 


1775 


1802 


1828 


1855 


1882 


27 


3.74 


1.3 1909 


1935 


1962 


1989 


2015 


2042 


2069 


2096 


2122 


2149 


27 


3.75 


1.3 2176 


2202 


2229 


2256 


2282 


2309 


2335 


2362 


2389 


2415 


27 


3.76 


1.3 2442 


2468 


2495 


2522 


2548 


2575 


2601 


2628 


2654 


2681 


27 


3.77 


1.3 2708 


2734 


2761 


2787 


2814 


2840 


2867 


2893 


2919 


2946 


27-26 


3.78 


1.3 2972 


2999 


3025 


3052 


3078 


3105 


3131 


3157 


3184 


3210 


26 


3.79 


1.3 3237 


3263 


3289 


3316 


3342 


3368 


3395 


3421 


3447 


3474 


26 


3.80 


1.3 3500 


3526 


3553 


3579 


3605 


3632 


3658 


3684 


3710 


3737 


26 


3.81 


1.3 3763 


3789 


3815 


3842 


3868 


3894 


3920 


3946 


3973 


3999 


26 


3.82 


1.3 4025 


4051 


4077 


4104 


4130 


4156 


4182 


4208 


4234 


4260 


26 


3.83 


1.3 4286 


4313 


4339 


4365 


4391 


4417 


4443 


4469 


4495 


4521 


26 


3.84 


1.3 4547 


4573 


4599 


4625 


4651 


4677 


4703 


4729 


4755 


4781 


26 


3.85 


1.3 4807 


4833 


4859 


4885 


4911 


4937 


4963 


4989 


5015 


5041 


26 


3.86 


1.3 5067 


5093 


5119 


5144 


5170 


5196 


5222 


5248 


5274 


5300 


26 


3.87 


1.3 5325 


5351 


5377 


5403 


5429 


5455 


5480 


5506 


5532 


5558 


26 


3.88 


1.3 5584 


5609 


5635 


5661 


5687 


5712 


5738 


5764 


5789 


5815 


26 


3.89 


1.3 5841 


5867 


5892 


5918 


5944 


5969 


5995 


6021 


6046 


6072 


26 


3.90 


1.3 6098 


6123 


6149 


6175 


6200 


6226 


6251 


6277 


6303 


6328 


26 


3.91 


1.3 6354 


6379 


6405 


6430 


6456 


6481 


6507 


6533 


6558 


6584 


26 


3.92 


1.3 6609 


6635 


6660 


6686 


6711 


6737 


6762 


6788 


6813 


6838 


26-25 


3.93 


1.3 6864 


6889 


6915 


6940 


6966 


6991 


7016 


7042 


7067 


7093 


25 


3.94 


1.3 7118 


7143 


7169 


7194 


7220 


7245 


7270 


7296 


7321 


7346 


25 


3.95 


1.3 7372 


7397 


7422 


7447 


7473 


7498 


7523 


7549 


7574 


7599 


25 


3.96 


1.3 7624 


7650 


7675 


7700 


7725 


7751 


7776 


7801 


7826 


7851 


25 


3.97 


1.3 7877 


7902 


7927 


7952 


7977 


8002 


8028 


8053 


8078 


8103 


25 


3.98 


1.3 8128 


81-13 


8178 


8204 


8229 


8254 


8279 


8304 


8329 


8354 


25 


3.99 


1.3 8379 


8404 


8429 


8454 


8479 


8504 


8529 


8554 


8579 


8604 


25 


4.00 


1.3 8629 


8654 


8679 


8704 


8729 


8754 


8779 


8804 


8829 


8854 


25 







1 


2 


3 


4 


5 


6 


7 


8 


9 





136 



TABLES. 
Five-Place Natural Logarithms. 



No. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 


4.00 


1.3 8629 


8654 


8679 


8704 


8729 


8754 


8779 


8804 


8829 


8854 


25 


4.01 


1.3 8879 


8904 


8929 


8954 


8979 


9004 


9029 


9054 


9078 


9103 


25 


4.02 


1.3 9128 


9153 


9178 


9203 


9228 


9252 


9277 


9302 


9327 


9352 


25 


4.03 


1.3 9377 


9401 


9426 


9451 


9476 


9501 


9525 


9550 


9575 


9600 


25 


4.04 


1.3 9624 


9649 


9674 


9699 


9723 


9748 


9773 


9798 


9822 


9847 


25 


4.0.S 


1.3 9872 


9896 


9921 


9946 


9970 


9995 


*0020 


0044 


0069 


0094 


25 


4.06 


1.4 0118 


0143 


0168 


0192 


0217 


0241 


0266 


0291 


0315 


0340 


25 


4.07 


1.4 0364 


0389 


0413 


0438 


04b3 


0487 


0512 


0536 


0561 


0585 


25 


4.08 


1.4 0610 


0634 


0659 


0683 


0708 


0732 


0757 


0781 


0806 


0830 


25-24 


4.09 


1.4 0854 


0879 


0903 


0928 


0952 


0977 


1001 


1025 


1050 


1074 


24 


4.10 


1.4 1099 


1123 


1147 


1172 


1196 


1221 


1245 


1269 


1294 


1318 


24 


4.11 


1.4 1342 


1367 


1391 


1415 


1440 


1464 


1488 


1512 


1537 


1561 


24 


4.12 


1.4 1585 


1610 


1634 


1658 


1682 


1707 


1731 


1755 


1779 


1804 


24 


4.13 


1.4 1828 


1852 


1876 


1900 


1925 


1949 


1973 


1997 


2021 


2045 


24 


4.14 


1.4 2070 


2094 


2118 


2142 


2166 


2190 


2214 


2239 


2263 


2287 


24 


4.15 


1.4 2311 


2335 


2359 


2383 


2407 


2431 


2455 


2479 


2503 


2527 


24 


4.16 


1.4 2552 


2576 


2600 


2624 


2648 


2672 


2696 


2720 


2744 


2768 


24 


4.17 


1.4 2792 


2816 


2840 


2864 


2887 


2911 


2935 


2959 


2983 


3007 


24 


4.18 


1.4 3031 


3055 


3079 


3103 


3127 


3151 


3175 


3198 


3222 


3246 


24 


4.19 


1.4 3270 


3294 


3318 


3342 


3365 


3389 


3413 


3437 


3461 


3485 


24 


4.20 


1.4 3508 


3532 


3556 


3580 


3604 


3627 


3651 


3675 


3699 


3723 


24 


4.21 


1.4 3746 


3770 


3794 


3817 


3841 


3865 


3889 


3912 


3936 


3960 


24 


4.22 


1.4 3984 


4007 


4031 


4055 


4078 


4102 


4126 


4149 


4173 


4197 


24 


4.23 


1.4 4220 


4244 


4267 


4291 


4315 


4338 


4362 


4386 


4409 


4433 


24 


4.24 


1.4 4456 


4480 


4503 


4527 


4551 


4574 


4598 


4621 


4645 


4668 


24 


4.25 


1.4 4692 


4715 


4739 


4762 


4786 


4809 


4833 


4856 


4880 


4903 


24-23 


4.26 


1.4 4927 


4950 


4974 


4997 


5021 


5044 


5068 


5091 


5115 


5138 


23 


4.27 


1.4 5161 


5185 


5208 


5232 


5255 


5278 


5302 


5325 


5349 


5372 


23 


4.28 


1.4 5395 


5419 


5442 


5465 


5489 


5512 


5535 


5559 


5582 


5605 


23 


4.29 


1.4 5629 


5652 


5675 


5699 


5722 


5745 


5768 


5792 


5815 


5838 


23 


4.30 


1.4 5862 


5885 


5908 


5931 


5954 


5978 


6001 


6024 


6047 


6071 


23 


4.31 


1.4 6094 


6117 


6140 


6163 


6187 


6210 


6233 


6256 


6279 


6302 


23 


4.32 


1.4 6326 


6349 


6372 


6395 


6418 


6441 


6464 


6487 


6511 


6534 


23 


4.33 


1.4 6557 


6580 


6603 


6626 


6649 


6672 


6695 


6718 


6741 


6764 


23 


4.34 


1.4 6787 


6810 


6834 


6857 


6880 


6903 


6926 


6949 


6972 


6995 


23 


4.35 


1.4 7018 


7041 


7064 


7087 


7109 


7132 


7155 


7178 


7201 


7224 


23 


4.36 


1.4 7247 


7270 


7293 


7316 


7339 


7362 


7385 


7408 


7431 


7453 


23 


4.37 


1.4 7476 


7499 


7522 


7545 


7568 


7591 


7614 


7636 


7659 


7682 


23 


4.38 


1.4 7705 


7728 


7751 


7773 


7796 


7819 


7842 


7865 


7887 


7910 


23 


4.39 


1.4 7933 


7956 


7978 


8001 


8024 


8047 


8070 


8092 


8115 


8138 


23 


4.40 


1.4 8160 


8183 


8206 


8229 


8251 


8274 


8297 


8319 


8342 


8365 


23 


4.41 


1.4 8387 


8410 


8433 


8455 


8478 


8501 


8523 


8546 


8569 


8591 


23 


4.42 


1.4 8614 


8637 


8659 


8682 


8704 


8727 


8750 


8772 


8795 


8817 


23 


4.43 


1.4 8840 


8863 


8885 


8908 


8930 


8953 


8975 


8998 


9020 


9043 


23 


4.44 


1.4 9065 


9088 


9110 


9133 


9155 


9178 


9200 


9223 


9245 


9268 


23 


4.45 


1.4 9290 


9313 


9335 


9358 


9380 


9403 


9425 


9448 


9470 


9492 


23-22 


4.46 


1.4 9515 


9537 


9560 


9582 


9605 


9627 


9649 


9672 


9694 


9716 


22 


4.47 


1.4 9739 


9761 


9784 


9806 


9828 


9851 


9873 


9895 


9918 


9940 


22 


4.48 


1.4 9962 


9985 


*0007 


0029 


0052 


0074 


0096 


0118 


0141 


0163 


22 


4.49 


1.5 0185 


0208 


0230 


0252 


0274 


0297 


0319 


0341 


0363 


0386 


22 


4.50 


1.5 0408 


0430 


0452 


0474 


0497 


0519 


0541 


0563 


0585 


0608 


22 







1 


2 


3 


4 


5 


6 


7 


8 


9 





TABLES. 
Five-Place Natural Logarithms. 



137 



No. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 


4.50 


1.5 0408 


0430 


0452 


0474 


0497 


0519 


0541 


0563 


0585 


0608 


22 


4.51 


1.5 0630 


0652 


0674 


0696 


0718 


0741 


0763 


0785 


0807 


0829 


22 


4.52 


1.5 0851 


0873 


0895 


0918 


0940 


0962 


0984 


1006 


1028 


1050 


22 


4.53 


1.5 1072 


1094 


1116 


1138 


1160 


1183 


1205 


1227 


1249 


1271 


22 


4.54 


1.5 1293 


1315 


1337 


1359 


1381 


1403 


1425 


1447 


1469 


1491 


22 


4.55 


1.5 1513 


1535 


1557 


1579 


1601 


1623 


1645 


1666 


1688 


1710 


22 


4.56 


1.5 1732 


1754 


1776 


1798 


1820 


1842 


1864 


1886 


1908 


1929 


22 


4.57 


1.5 1951 


1973 


1995 


2017 


2039 


2061 


2083 


2104 


2126 


2148 


22 


4.58 


1.5 2170 


2192 


2214 


2235 


2257 


2279 


2301 


2323 


2344 


2366 


22 


4.59 


1.5 2388 


2410 


2432 


2453 


2475 


2497 


2519 


2540 


2562 


2584 


22 


4.60 


1.5 2606 


2627 


2649 


2671 


2693 


2714 


2736 


2758 


2779 


2801 


22 


4.61 


1.5 2823 


2844 


2866 


2888 


2910 


2931 


2953 


2975 


2996 


3018 


22 


4.62 


1.5 3039 


3061 


3083 


3104 


3126 


3148 


3169 


3191 


3212 


3234 


22 


4.63 


1.5 3256 


3277 


3299 


3320 


3342 


3364 


3385 


3407 


3428 


3450 


22 


4.64 


1.5 3471 


3493 


3515 


3536 


3558 


3579 


3601 


3622 


3644 


3665 


22 


4.65 


1.5 3687 


3708 


3730 


3751 


3773 


3794 


3816 


3837 


3859 


3880 


22-21 


4.66 


1.5 3902 


3923 


3944 


3966 


3987 


4009 


4030 


4052 


4073 


4094 


21 


4.67 


1.5 4116 


4137 


4159 


4180 


4202 


4223 


4244 


4266 


4287 


4308 


21 


4.68 


1.5 4330 


4351 


4373 


4394 


4415 


4437 


4458 


4479 


4501 


4522 


21 


4.69 


1.5 4543 


4565 


4586 


4607 


4629 


4650 


4671 


4692 


4714 


4735 


21 


4.70 


1.5 4756 


4778 


4799 


4820 


4841 


4863 


4884 


4905 


4926 


4948 


21 


4.71 


1.5 4969 


4990 


5011 


5032 


5054 


5075 


5096 


5117 


5138 


5160 


21 


4.72 


1.5 5181 


5202 


5223 


5244 


5266 


5287 


5308 


5329 


5350 


5371 


21 


4.73 


1.5 5393 


5414 


5435 


5456 


5477 


5498 


5519 


5540 


5562 


5583 


21 


4.74 


1.5 5604 


5625 


5646 


5667 


5688 


5709 


5730 


5751 


5772 


5793 


21 


4.75 


1.5 5814 


5836 


5857 


5878 


5899 


5920 


5941 


5962 


5983 


6004 


21 


4.76 


1.5 6025 


6046 


6067 


6088 


6109 


6130 


6151 


6172 


6193 


6214 


21 


4.77 


1.5 6235 


6256 


6277 


6298 


6318 


6339 


6360 


6381 


6402 


6423 


21 


4.78 


1.5 6444 


6465 


6486 


6507 


6528 


6549 


6569 


6590 


6611 


6632 


21 


4.79 


1.5 6653 


6674 


6695 


6716 


6737 


6757 


6778 


6799 


6820 


6841 


21 


4.80 


1.5 6862 


6882 


6903 


6924 


6945 


6966 


6987 


7007 


7028 


7049 


21 


4.81 


1.5 7070 


7090 


7111 


7132 


7153 


7174 


7194 


7215 


7236 


7257 


21 


4.82 


1.5 7277 


7298 


7319 


7340 


7360 


7381 


7402 


7423 


7443 


7464 


21 


4.83 


1.5 7485 


7505 


7526 


7547 


7567 


7588 


7609 


7629 


7650 


7671 


21 


4.84 


1.5 7691 


7712 


7733 


7753 


7774 


7795 


7815 


7836 


7857 


7877 


21 


4.85 


1.5 7898 


7918 


7939 


7960 


7980 


8001 


8022 


8042 


8063 


8083 


21 


4.86 


1.5 8104 


8124 


8145 


8166 


8186 


8207 


8227 


8248 


8268 


8289 


21 


4.87 


1.5 8309 


8330 


8350 


8371 


8391 


8412 


8433 


8453 


8474 


8494 


21-20 


4.88 


1.5 8515 


8535 


8555 


8576 


8596 


8617 


8637 


8658 


8678 


8699 


20 


4.89 


1.5 8719 


8740 


8760 


8781 


8S01 


8821 


8842 


8862 


8883 


8903 


20 


4.90 


1.5 8924 


8944 


8964 


8985 


9005 


9026 


9046 


9066 


9087 


9107 


20 


4.91 


1.5 9127 


9148 


9168 


9188 


9209 


9229 


9250 


9270 


9290 


9311 


20 


4.92 


1.5 9331 


9351 


9371 


9392 


9412 


9432 


9453 


9473 


9493 


9514 


20 


4.93 


1.5 9534 


955' 


9574 


9595 


9615 


9635 


9656 


9676 


9696 


9716 


20 


4.94 


1.5 9737 


9757 


9777 


9797 


9817 


9838 


9858 


9878 


9898 


9919 


20 


4.95 


1.5 9939 


9959 


9979 


9999 *0020 


0040 


0060 


OOSO 


0100 


0120 


20 


4.96 


1.6 0141 


0161 


0181 


0201 


0221 


0241 


0261 


028? 


0302 


0322 


20 


4.97 


1.6 0342 


0362 


0382 


0402 


0422 


0443 


0463 


0483 


0503 


0523 


20 


4.98 


1.6 0543 


0563 


0583 


0603 


0623 


0643 


0663 


0683 


0704 


0724 


20 


4.99 


1.6 0744 


0764 


0784 


0804 


0824 


0844 


0864 


0884 


0904 


0924 


20 


5>00 


1.6 0944 


0964 


0984 


1004 


1024 


1044 


1064 


1084 


1104 


1124 


20 







1 


2 


3 


4 


5 


6 


7 


8 


9 





138 



TABLES. 



Five-Place Natural Logarithms. 



No. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 


5.0 


1.6 0944 


1144 


1343 


1542 


1741 


1939 


2137 


2334 


2531 


2728 


200-196 


5.1 


1.6 2924 


3120 


3315 


3511 


3705 


3900 


4094 


4287 


44S1 


4673 


196-192 


5.2 


1.6 4866 


5058 


5250 


5441 


5632 


5823 


6013 


6203 


6393 


6582 


192-189 


5.3 


1.6 6771 


6959 


7147 


7335 


7523 


7710 


7896 


8083 


8269 


8455 


189-185 


5.4 


1.6 8640 


8825 


9010 


9194 


9378 


9562 


9745 


9928 *0111 


0293 


185-182 


5.5 


1.7 0475 


0656 


0838 


1019 


1199 


1380 


1560 


1740 


1919 


2098 


182-179 


5.6 


1.7 2277 


2455 


2633 


2811 


2988 


3166 


3342 


3519 


3695 


3871 


178-176 


5.7 


1.7 4047 


4222 


4397 


4572 


4746 


4920 


5094 


5267 


5440 


5613 


175-173 


5.8 


1.7 5786 


5958 


6130 


6302 


6473 


6644 


6815 


6985 


7156 


7326 


172-170 


5.9 


1.7 7495 


7665 


7834 


8002 


8171 


8339 


8507 


8675 


8842 


9009 


169-167 


6.0 


1.7 9176 


9342 


9509 


9675 


9840 


*0006 


0171 


0336 


0500 


0665 


167-164 


6.1 


1.8 0829 


0993 


1156 


1319 


1482 


1645 


1808 


1970 


2132 


2294 


164-161 


6.2 


1.8 2455 


2616 


2777 


2938 


3098 


3258 


3418 


3578 


3737 


3896 


161-159 


6.3 


1.8 4055 


4214 


4372 


4530 


4688 


4845 


5003 


5160 


5317 


5473 


159-156 


6.4 


1.8 5630 


5786 


5942 


6097 


6253 


6408 


6563 


6718 


6872 


7026 


156-154 


6.5 


1.8 7180 


7334 


7487 


7641 


7794 


7947 


8099 


8251 


8403 


8555 


154-152 


6.6 


1.8 8707 


8858 


9010 


9160 


931] 


9462 


9612 


9762 


9912 *0061 


151-149 


6.7 


1.9 0211 


0360 


0509 


0658 


0806 


0954 


1102 


1250 


1398 


1545 


149-147 


6.8 


1.9 1692 


1839 


1986 


2132 


2279 


2425 


2571 


2716 


2862 


3007 


147-145 


6.9 


1.9 3152 


3297 


3442 


3586 


3730 


3874 


4018 


4162 


4305 


4448 


145-143 


7.0 


1.9 4591 


4734 


4876 


5019 


5161 


5303 


5445 


5586 


5727 


5869 


143-141 


7.1 


1.9 6009 


6150 


6291 


6431 


6571 


6711 


6851 


6991 


7130 


7269 


141-139 


7.2 


1.9 7408 


7547 


7685 


7824 


7962 


8100 


8238 


8376 


8513 


8650 


139-137 


7.3 


1.9 8787 


8924 


9061 


9198 


9334 


9470 


9606 


9742 


9877 *0013 


137-135 


7.4 


2.0 0148 


0283 


0418 


0553 


0687 


0821 


0956 


1089 


1223 


1357 


135-133 


7.5 


2.0 1490 


1624 


1757 


1890 


2022 


2155 


2287 


2419 


2551 


2683 


133-132 


7.6 


2.0 2815 


2946 


3078 


3209 


3340 


3471 


3601 


3732 


3862 


3992 


131-130 


7.7 


2.0 4122 


4252 


4381 


4511 


4640 


4769 


4898 


5027 


5156 


5284 


130-128 


7.8 


2.0 5412 


5540 


5668 


5796 


5924 


6051 


6179 


6306 


6433 


6560 


128-127 


7.9 


2.0 6686 


6813 


6939 


7065 


7191 


7317 


7443 


7568 


7694 


7819 


127-125 


8.0 


2.0 7944 


8069 


8194 


8318 


8443 


8567 


8691 


8815 


8939 


9063 


125-124 


8.1 


2.0 9186 


9310 


9433 


9556 


9679 


9802 


9924 *0047 


0169 


0291 


123-122 


8.2 


2.1 0413 


0535 


0657 


0779 


0900 


1021 


1142 


1263 


1384 


1505 


122-121 


8.3 


2.1 1626 


1746 


1866 


1986 


2106 


2226 


2346 


2465 


2585 


2704 


120-119 


8.4 


2.1 2823 


2942 


3061 


3180 


3298 


3417 


3535 


3653 


3771 


3889 


119-118 


8.5 


2.1 4007 


4124 


4242 


4359 


4476 


4593 


4710 


4827 


4943 


5060 


118-116 


8.6 


2.1 5176 


5292 


5409 


5524 


5640 


5756 


5871 


5987 


6102 


6217 


116-115 


8.7 


2.1 6332 


6447 


6562 


6677 


6791 


6905 


7020 


7134 


7248 


7361 


115-114 


8.8 


2.1 7475 


7589 


7702 


7816 


7929 


8042 


8155 


8267 


8380 


8493 


114-112 


8.9 


2.1 8605 
2.1 9722 


8717 
9834 


8830 8942 
9944 *0055 


9054 
0166 


9165 


9277 


9389 


9500 


9611 


112-111 


9.0 


0276 


0387 


0497 


0607 


0717 


111-110 


9.1 


2.2 0827 


0937 


1047 


1157 


1266 


1375 


1485 


1594 


1703 


1812 


110-109 


9.2 


2.2 1920 


2029 


2138 


2246 


2354 


2462 


2570 


2678 


2786 


2894 


109-108 


9.3 


2.2 3001 


3109 


3216 


3324 


3431 


3538 


3645 


3751 


3858 


3965 


107-106 


9.4 


2.2 4071 


4177 


4284 


4390 


4496 


4601 


4707 


4813 


4918 


5024 


106-105 


9.5 


2.2 5129 


5234 


5339 


5444 


5549 


5654 


5759 


5863 


5968 


6072 


105-104 


9.6 


2.2 6176 


6280 


6384 


6488 


6592 


6696 


6799 


6903 


7006 


7109 


104-103 


9.7 


2.2 7213 


7316 


7419 


7521 


7624 


7727 


7829 


7932 


8034 


8136 


103-102 


9.8 


2.2 8238 


8340 


8442 


8544 


8646 


8747 


8849 


8950 


9051 


9152 


102-101 


9.9 


2.2 9253 


9354 


9455 


9556 


9657 


9757 


9858 


9958 *0058 


0158 


101-100 


10.0 


2.3 0259 


0358 


0458 


0558 


0658 


0757 


0857 


0956 


1055 


1154 


100-99 







1 


2 


3 


4 


5 


6 


7 


8 


9 





TABLES. 139 

The Natural Logarithms (each increased by 10.) of Numbers between 0.00 and 0.99. 



No. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


0.0 




5.395 


6.088 


6.493 


6.781 


7.004 


7.187 


7.341 


7.474 


7.592 


0.1 


7.697 


7.793 


1880 


7.960 


8.034 


8.103 


8.167 


8.228 


8.285 


8.339 


0.2 


8.391 


8.439 


8.486 


8.530 


8.573 


8.614 


8.653 


8.691 


8.727 


8.762 


0.3 


8.796 


8.829 


8.861 


8.891 


8.921 


8.950 


8.978 


9.006 


9.032 


9.058 


0.4 


9.084 


9.10S 


9.132 


9.156 


9.179 


9.201 


9.223 


9.245 


9.266 


9.287 


0.5 


9.307 


9.327 


9.346 


9.365 


9.384 


9.402 


9.420 


9.438 


9.455 


9.472 


0.6 


9.489 


9.506 


9.522 


9.538 


9.554 


9.569 


9.584 


9.600 


9.614 


9.629 


0.7 


9.643 


9.658 


9.671 


9.685 


9.699 


9.712 


9.726 


9.739 


9.752 


9.764 


0.8 


9.777 


9.789 


9.802 


9.814 


9.826 


9.837 


9.849 


9.861 


9.872 


9.883 


0.9 


9.895 


9.906 


9.917 


9.927 


9.938 


9.949 


9.959 


9.970 


9.980 


9.990 



Note : loggX = logioX • lege 10 =: (2.30259) logio x. 



The Natural Logarithms of Whole Numbers from 10 to 209. 



No. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


1 


2.3026 


3979 


4849 


5649 


6391 


7080 


7726 


8332 


8904 


9444 


2 


2.9957 


*0445 


0910 


1355 


1781 


2189 


2581 


2958 


3322 


3673 


3 


3.4012 


4340 


4657 


4965 


5264 


5553 


5835 


6109 


6376 


6636 


4 


3.6889 


7136 


7377 


7612 


7842 


8067 


8286 


8501 


8712 


8918 


5 


3.9120 


9318 


9512 


9703 


9890 


*0073 


0254 


0431 


0604 


0775 


6 


4.0943 


1109 


1271 


1431 


1589 


1744 


1897 


2047 


2195 


2341 


7 


4.2485 


2627 


2767 


2905 


3041 


3175 


3307 


3438 


3567 


3694 


8 


4.3820 


3944 


4067 


4188 


4308 


4427 


4543 


4659 


4773 


4886 


9 


4.4998 


5109 


5218 


5326 


5433 


5539 


5643 


5747 


5850 


5951 


10 


4.6052 


6151 


6250 


6347 


6444 


6540 


6634 


6728 


6821 


6913 


11 


4.7005 


7095 


7185 


7274 


7362 


7449 


7536 


7622 


7707 


7791 


12 


4.7875 


7958 


8040 


8122 


8203 


8283 


8363 


8442 


8520 


8598 


13 


4.8675 


8752 


8828 


8903 


8978 


9053 


9127 


9200 


9273 


9345 


14 


4.9416 


94SS 


9558 


9628 


9698 


9767 


9836 


9904 


9972 


*0039 


15 


5.0106 


0173 


0239 


0304 


0370 


0434 


0499 


0562 


0626 


0689 


16 


5.0752 


0814 


0876 


0938 


0999 


1059 


1120 


1180 


1240 


1299 


17 


5.1358 


P17 


1475 


1533 


1591 


1648 


1705 


1762 


1818 


1874 


18 


5.1930 


1985 


2040 


2095 


2149 


2204 


2257 


2311 


2364 


2417 


19 


5.2470 


2523 


2575 


2627 


2679 


2730 


2781 


2832 


2883 


2933 


20 


5.2983 


3033 


3083 


3132 


3181 


3230 


3279 


3327 


3375 


3423 



Note : loge 10 = 2,30258509. 



lege 100 = 4.60517019. 



140 



TABLES. 



The Common Logarithms of r (n) for Values of n between 1 and 2. 
r(n)= j x"-i-e-^dx= j log- dx. 



n 


9^ 

o 


n 


o 

ho 

O 

r— » 


n 


3^ 

o 


n 


3; 

2 

0^ 


n 


3; 

bO 


1.01 


1.9975 


1.21 


T.9617 


1.41 


1.9478 


1.61 


1.9517 


1.81 


1.9704 


1.02 


1.9951 


1.22 


1.9605 


1.42 


1.9476 


1.62 


1.9523 


1.82 


1.9717 


1.03 


1.9928 


1.23 


1.9594 


1.43 


1.9475 


1.63 


1.9529 


1.83 


1.9730 


1.04 


1.9905 


1.24 


1.9583 


1.44 


1.9473 


1.64 


1.9536 


1.84 


1.9743 


1.05 


1.9883 


1.25 


1.9573 


1.45 


1.9473 


1.65 


1.9543 


1.85 


1.9757 


1.06 


T.9862 


1.26 


1.9564 


1.46 


1.9472 


1.66 


1.9550 


1.86 


1.9771 


1.07 


T.9841 


1.27 


T.9554 


1.47 


1.9473 


1.67 


1.9558 


1.87 


1.9786 


l.OS 


1.9821 


1.28 


1.9546 


1.48 


1.9473 


1.68 


T.9566 


1.88 


1.9800 


1.09 


1.9802 


1.29 


1.9538 


1.49 


1.9474 


1.69 


1.9575 


1.89 


1.9815 


1.10 


1.9783 


1.30 


1.9530 


1.50 


1.9475 


1.70 


T.9584 


1.90 


1.9831 


1.11 


1.9765 


1.31 


T.9523 


1.51 


1.9477 


1.71 


1.9593 


1.91 


T.9846 


1.12 


1.9748 


1.32 


1.9516 


1.52 


1.9479 


1.72 


1.9603 


1.92 


1.9862 


1.13 


1.9731 


1.33 


1.9510 


1.53 


1.9482 


1.73 


1.9613 


1.93 


1.9878 


1.14 


1.9715 


1.34 


T.9505 


1.54 


1.9485 


1.74 


T.9623 


1.94 


1.9895 


1.15 


1.9699 


1.35 


1.9500 


1.55 


1.9488 


1.75 


1.9633 


1.95 


1.9912 


1.16 


1.9684 


1.36 


1.9495 


1.56 


1.9492 


1.76 


1.9644 


1.96 


T.9929 


1.17 


1.9669 


1.37 


1.9491 


1.57 


1.9496 


1.77 


T.9656 


1.97, 


1.9946 


1.18 


1.9655 


1.38 


1.9487 


1.5S 


1.9501 


1.78 


T.9667 


1.98 


1.9964 


1.19 


1.9642 


1.39 


1.9483 


1.59 


1.9506 


1.79 


1.9679 


1.99 


1.9982 


1.20 


1.9629 


1.40 


1.9481 


1.60 


1.9511 


1.80 


1.9691 


200 


0.0000 



r(2 + i) = z-r(2), z>i. 



TABLES. 



141 



NATURAL TRIGONOMETRIC FUNCTIONS. 



Angle. 


Sin. 


Csc. 


Tan. 


Ctn, 


Sec. 


Cos. 




0° 


0.000 


00 


0.000 


00 


1.000 


1.000 


90° 


1 


0.017 


57.30 


0.017 


57.29 


1.000 


1.000 


89 


2 


0.035 


28.65 


0.035 


28.64 


1.001 


0.999 


88 


3 


0.052 


19.11 


0.052 


19.08 


1.001 


0.999 


87 


4 


0.070 


14.34 


0.070 


14.30 


1.002 


0.998 


86 


5° 


0.0S7 


11.47 


0.0S7 


11.43 


1.004 


0.996 


85° 


6 


0.105 


9.567 


0.105 


9.514 


1.006 


0.995 


84 


7 


0.122 


8.206 


0.123 


8.144 


1.008 


0.993 


83 


8 


0.139 


7.185 


0.141 


7.115 


1.010 


0.990 


82 


9 


0.156 


6.392 


0.158 


6.314 


1.012 


0.988 


81 


10° 


0.174 


5.759 


0.176 


5.671 


1.015 


0.985 


80° 


11 


0.191 


5.241 


0.194 


5.145 


1.019 


0.982 


79 


12 


0.208 


4.810 


0.213 


4.705 


1.022 


0.978 


78 


13 


0.225 


4.445 


0.231 


4.331 


1.026 


0.974 


77 


14 


0.242 


4.134 


0.249 


4.011 


1.031 


0.970 


76 


15° 


0.259 


3.864 


0.268 


3.732 


1.035 


0.966 


75° 


16 


0.276 


3.628 


0.287 


3.487 


1.040 


0.961 


74 


17 


0.292 


3.420 


0.306 


3.271 


1.046 


0.956 


73 


18 


0.309 


3.236 


0.325 


3.078 


1.051 


0.951 


72 


19 


0.326 


3.072 


0.344 


2.904 


1.058 


0.946 


71 


20° 


0.342 


2.924 


0.364 


2.747 


1.064 


0.940 


70° 


21 


0.358 


2.790 


0.384 


2.605 


1.071 


0.934 


69 


22 


0.375 


2.669 


0.404 


2.475 


1.079 


0.927 


68 


23 


0.391 


2.559 


0.424 


2.356 


1.086 


0.921 


67 


24 


0.407 


2.459 


0.445 


2.246 


1.095 


0.914 


66 


25^ 


0.423 


2.366 


0.466 


2.145 


1.103 


0.906 


65° 


26 


0.438 


2.281 


0.488 


2.050 


1.113 


0.899 


64 


27 


0.454 


2.203 


0.5 JO 


1.963 


1.122 


0.891 


63 


28 


0.469 


2.130 


0.532 


1.881 


1.133 


0.883 


62 


29 


0.485 


2.063 


0.554 


1.804 


1.143 


0.875 


61 


30" 


0.500 


2.000 


0.577 


1.732 


1.155 


0.866 


60° 


31 


0.515 


1.942 


0.601 


1.664 


1.167 


0.857 


59 


32 


0.530 


1.8S7 


0.625 


1.600 


1.179 


0.848 


58 


33 


0.545 


1.836 


0.649 


1.540 


1.192 


0.839 


57 


34 


0.559 


1.788 


0.675 


1.483 


1.206 


0.829 


56 


35° 


0.574 


1.743 


0.700 


1.428 


1.221 


0.819 


55° 


36.. 
'37 ;^ 
S8 


0.588 


1.701 

1 £.ilC%- ■■■ 


0.727 


1.376 

-1.327 

1.280 


1.236 
1.252 
1.269 


0.809 
0.799 
0.788 


54 
53 
52 


0.602 
0.616 


1.662 
1.624 


0.754- 
0.781 


39 


0.629 


1.589 


0.810 


1.235 


1.287 


0.777 


51 


40° 


0.643 


1.556 


0.839 


1.192 


1.305 


0.766 


50° ' 


41 


0.656 


1.524 


0.869 


1.150 


1.325 


0.755 


49 


42 


0.669 


1.494 


0.900 


1.111 


1.346 


0.743 


48 


43 


0.682 


1.466 


0.933 


1.072 


1.367 


0.731 


47 


44 


0.695 


1.440 


0.966 


1.036 


1.390 


0.719 


46 


45° 


0.707 


1.414 


1.000 


1.000 


1.414 


0.707 


45° 




Cos. 


Sec. 


Ctn. 


Tan. 


Csc. 


Sin. 


Angle. 



142 



TABLES. 
Logarithms. 



N 





1 


2 


3 


4 5 


6 


7 


8 


9 


P.P. 

1.2- 3. 4- 5 


lO 


0000 


0043 


0086 


0128 


0170 


0212 


0253 


0294 


0334 


0374 


4- 8-12.17.21 


11 


0414 


0453 


0492 


0531 


0569 


0607 


0645 


0682 


0719 


0755 


4 8.11.10-19 


12 


0792 


0828 


0864 


0899 


0934 


0969 


1004 


1038 


1072 


1106 


3- 7-10. 14-17 


13 


1139 


1173 


1206 


1239 


1271 


1303 


1335 


1367 


1399 


1430 


3- 610.13.16 


14 
15 


1461 


1492 


1523 


1553 


1584 


1614 


1044 


1673 


1703 


1732 


3. 6. 9.12.15 


1761 


1790 


1818 


1847 


1875 


1903 


1931 


1959 


1987 


2014 


3. 6. 8-11.14 


16 


2041 


2068 


2095 


2122 


2148 


2175 


2201 


2227 


2253 


2279 


3. 5. 81113 


17 


2304 


2330 


2355 


2380 


2405 


2430 


2455 


2480 


2504 


2529 


2. 5- 7-10-12 


18 


2553 


2577 


2601 


2625 


2648 


2672 


2695 


2718 


2742 


2765 


2- 5- 7. 9-12 


19 
20 


2788 


2810 


2833 


2856 


2878 


2900 


2923 


2945 


2967 


2989 


2. 4. 7. 911 


3010 


3032 


3054 


3075 


3096 


3118 


3139 


3160 


3181 


3201 


2- 4- 6- 8 11 


21 


3222 


3243 


3263 


3284 


3304 


3324 


3345 


3365 


3385 


3404 


2. 4. 6. 810 


22 


3424 


3444 


3464 


3483 


3502 


3522 


3541 


3560 


3579 


3598 


2- 4. 6- 8.10 


23 


3617 


3636 


3655 


3674 


3692 


3711 


3729 


3747 


3766 


3784 


2. 4. 5- 7- 9 


24 
25 


3802 


3820 


3838 


3856 


3874 


3892 


3909 


3927 


3945 


3962 


2- 4. 5- 7. 9 


3979 


3997 


4014 


4031 


4048 


4065 


4082 


4099 


4110 


4133 


2 3. 5. 7. 9 


26 


4150 


4166 


4183 


4200 


4216 


4232 


4249 


4265 


4281 


4298 


2. 3. 5. 7. 8 


27 


4314 


4330 


4346 


4362 


4378 


4393 


4409 


4425 


4440 


4456 


2. 3. 5. 6- 8 


28 


4472 


4487 


4502 


4518 


4533 


4548 


4564 


4579 


4594 


4609 


2- 3. 5. 6. 8 


29 
30 


4624 


4639 


4654 


4669 


4683 


4698 


4713 


4728 


4742 


4757 


1. 3- 4. 6. 7 


4771 


4786 


4800 


4814 


4829 


4843 


4857 


4871 


4886 


4900 


1- 3. 4. 6. 7 


31 


4914 


4928 


4942 


4955 


4969 


4983 


4997 


5011 


5024 


5038 


1- 3- 4. 6- 7 


32 


5051 


5065 


5079 


5092 


5105 


5119 


5132 


5145 


5159 


5172 


1. 3. 4- 5- 7 


33 


5185 


5198 


5211 


5224 


5237 


5250 


5263 


5276 


5289 


5302 


1- 3. 4- 5. 6 


34 
35 


5315 


5328 


5340 


5353 


5366 


5378 


5391 


5403 


5416 


5428 


1. 3- 4- 5- 6 


5441 


5453 


5465 


5478 


5490 


5502 


5514 


5527 


5539 


5551 


1- 2- 4- 5- 6 


36 


5563 


5575 


5587 


5599 


5611 


5623 


5635 


5647 


5658 


5670 


1- 2- 4. 5. 6 


37 


5682 


5694 


5705 


5717 


5729 


5740 


5752 


5763 


5775 


5786 


1-2. 3- 5. 6 


38 


5798 


5809 


5821 


5832 


58i3 


5855 


5866 


5877 


5888 


5899 


1- 2- 3. 5. 6 


39 
40 


5911 


5922 


5933 


5944 


5955 


5966 


5977 


5988 


5999 


6010 


1- 2- 3. 4- 6 


6021 


6031 


6042 


6053 


6064 


6075 


6085 


6096 


6107 


6117 


1- 2- 3. 4. 5 


41 


6128 


6138 


6149 


6160 


6170 


6180 


6191 


6201 


6212 


6222 


1- 2. 3. 4. 5 


42 


6232 


6243 


6253 


6263 


6274 


6284 


6294 


6304 


6314 


6325 


1- 2- 3- 4. 5 


43 


6335 


6345 


6355 


6365 


6375 


6385 


6395 


6405 


6415 


6425 


1- 2. 3. 4- 5 


44 
45 


6435 


6444 


6454 


6464 


6474 


6484 


6493 


6503 


6513 


6522 


1. 2. 3- 4- 5 


6532 


6542 


6551 


65G1 


6571 


6580 


6590 


6599 


6809 


6618 


1. 2- 3. 4. 5 


46 


6628 


6637 


6646 


6656 


6665 


6675 


6684 


6693 


6702 


6712 


1. 2- 3- 4- 5 


47 


6721 


6730 


6739 


6749 


6758 


6767 


6776 


6785 


6794 


6803 


1. 2. 3. 4. 5 


48 


6812 


6821 


6830 


6839 


6848 


6857 


6366 


6875 


6884 


6893 


1. 2. 3 4. 4 


49 
50 


6902 


6911 


6920 


6928 


6937 


6946 


6955 


6964 


6972 


6981 


1- 2- 3- 4- 4 


6990 


6998 


7007 


7016 


7024 


7033 


7042 


7050 


7059 


7067 


1- 2. 3. 3. 4 


51 


7076 


7084 


7093 


7101 


7110 


7118 


7126 


7135 


7143 


7152 


1- 2- 3. 3. 4 


52 


7160 


7168 


7177 


7185 


7193 


7202 


7210 


7218 


7226 


7235 


1- 2- 2- 3. 4 


53 


7243 


7251 


7259 


7267 


7275 


7284 


7292 


7300 


7308 


7316 


1- 2. 2- 3. 4 


54 


7324 


7332 


7340 


7348 


7356 


7364 


7372 


7380 


7388 


7396 


1. 2- 2. 3. 4 



NoTK. — This page and the three that follow it are taken from the Mathematical 
Tables of Prof. J. M. Peirce, published by ^Messrs. Ginn & Co. 



TABLES. 
Logarithms. 



143 



[ 

N 





12 3 


4 5 


6 


7 


8 


9 


P P. 


















1. ^ 


• 3- 4. 5 


55 


7404 


7412 7419 7427 


7435 


7443 


7451 


7459 


7466 


7474 


J. 2 


. 2. 3- 4 


56 


7482 


7490 7497 7505 


7513 


7520 


7528 


7536 


7543 


7551 


1-2 


.23-4 


57 


7559 


7566 7574 7582 


7589 


7597 


7604 


7612 


7619 


7627 


1. 2 


• 2. 3. 4 


58 


7634 


7642 7649 7657 


7664 


7672 


7679 


7686 


7694 


7701 




2-3. 4 


59 
60 


7709 


7716 7723 7731 


7738 


7745 


7752 


7760 


7767 


7774 




2. 3 4 


7782 


7789 7796 7803 


7810 


7818 


7825 


7832 


7839 


7846 




2-3.4 


61 


7853 


7860 7868 7875 


7882 


7889 


7896 


7903 


7910 


7917 




2.3-4 


62 


7924 


7931 7938 7945 


7952 


7959 


7966 


7973 


7980 


7987 




2.3.3 


63 


7993 


8000 8007 8014 


8021 


8028 


8035 


8041 


8048 


8055 




2.3- 3 


64 
65 


8062 


8069 8075 8082 


8089 


8096 


8102 


8109 


8116 


8122 




2.3.3 


8129 


8136 8142 8149 


8156 


8162 


8169 


8176 


8182 


8189 




2-3.3 


66 


8195 


8202 8200 8215 


8222 


8228 


8235 


8241 


8248 


8254 




2.3.3 


67 


8261 


8267 8274 8280 


8287 


8293 


8299 


8306 


8312 


8319 




2.3.3 


68 


8325 


8331 8338 8344 


8351 


8357 


8363 


8370 


8376 


8382 




2.3. 3 


69 
70 


8388 


8395 8401 8407 


8414 


8420 


8426 


8432 


8439 


8445 




2. 3.3 


8451 


8457 8463 8470 


8476 


8482 


8488 


8494 


8500 


8506 




2.2.3 


71 


8513 


8519 8525 8531 


8537 


8543 


8549 


8555 


8561 


8567 




2-2.3 


72 


8573 


8579 8585 8591 


8597 


8603 


8609 


8615 


8621 


8627 




2.2.3 


73 


8633 


8639 8645 8651 


8657 


8663 


8669 


8675 


8681 


8686 




2. 2-3 


74 
75 


8692 


8698 8704 8710 


8716 


8722 


8727 


8733 


8739 


8745 




2-2.3 


8751 


8756 8762 8768 


8774 


8779 


8785 


8791 


8797 


8802 




2.2.3 


76 


8808 


8814 8820 8825 


8831 


8837 


8842 


8848 


8854 


8859 




2-2-3 


77 


8865 


8871 8876 8882 


8887 


8893 


8899 


8904 


8910 


8915 




2.2.3 


78 


8921 


8927 8932 8938 


8943 


8949 


8954 


8960 


8965 


8971 




2.2.3 


79 
80 


8976 


8982 8987 8993 


8998 


9004 


9009 


9015 


9020 


9025 




2- 2. 3 


9031 


9036 9042 9047 


9053 


9058 


9063 


9069 


9074 


9079 




2-2.3 


81 


9085 


9090 9096 9101 


9106 


9112 


9117 


9122 


9128 


9133 




2.2.3 


82 


9138 


9143 9149 9154 


9159 


9165 


9170 


9175 


9180 


9186 




2.2-3 


83 


9191 


9196 9201 9206 


9212 


9217 


9222 


9227 


9232 


9238 




2.2-3 


84 
85 


9243 


9248 9253 9258 


9263 


9269 


9274 


9279 


9284 


9289 




2.2.3 


9294 


9299 9304 9309 


9315 


9320 


9325 


9330 


9335 


9340 




2 2.3 


86 


9345 


9350 9355 9360 


9365 


9370 


9375 


9380 


9385 


9390 




2-2.3 


87 


9395 


9400 9405 9410 


9415 


9420 


9425 


9430 


9435 


9440 


0. 1 


1.2.2 


88 


9445 


9450 9455 9460 


9465 


9469 


9474 


9479 


9484 


9489 


0.1 


1- 2-2 


89 
90 


9494 


9499 9504 9509 


9513 


9518 


9523 


9528 


9533 


9538 


0. 1 


1.2.2 


9542 


9547 9552 9557 


9562 


9566 


9571 


9576 


9581 


9586 


0- 1 


1.2.2 


91 


9590 


9595 9600 9605 


9609 


9614 


9619 


9624 


9628 


9633 


0. 1 


1.2.2 


92 


9638 


9643 9647 9652 


9657 


9661 


9666 


9671 


9675 


9680 


0- 1 


1.2-2 


93 


9685 


9689 9694 9699 


9703 


9708 


9713 


9717 


9722 


9727 


0- 1 


1.2.2 


94 
95 


9731 


9736 9741 9745 


9750 


9754 


9759 


9763 


9768 


9773 


1 


1. 2- 2 


9777 


9782 9786 9791 


9795 


9800 


9805 


9809 


9814 


9818 


0. 1 


1.2.2 


96 


9823 


9827 9832 9836 


9841 


9845 


9850 


9854 


9859 


9863 


0. 1 


1.2.2 


97 


9868 


9872 9877 9881 


9886 


9890 


9894 


9899 


9903 


9908 


0- 1 


1.2.2 


98 


9912 


9917 9921 9926 


9930 


9934 


9939 


9943 


9948 


9952 


0- 1 


. 1. 2. 2 


99 


9956 


9961 9965 9969 


9974 


9978 


9983 


9987 


9991 


9996 


0- 1 


1.2-2 



log !r= 0.49715- 



log e = 0.43429 - 



144 



TABLES. 

Logarithms. 



N 





T 


2 


3 4 5 


6 


7 


8 


9 10 


100 


0000 


0004 


0009 


0013 0017 


0022 


0026 


0030 


0035 


0039 


0043 


101 


0043 


0043 


0052 


0056 0060 


0065 


0069 


0073 


0077 


0082 


0086 


102 


0086 


0090 


0095 


0099 0103 


0107 


0111 


0116 


0120 


0124 


0128 


103 


0128 


0133 


0137 


0141 0145 


0149 


0154 


0158 


0162 


0166 


0170 


104 
105 


0170 


0175 


0179 


0183 0187 


0191 


0195 


0199 


0204 


0208 


0212 


0212 


0216 


0220 


0224 0228 


0233 


0237 


0241 


0245 


0249 


0253 


106 


0253 


0257 


0261 


0265 0269 


0273 


0278 


0282 


0286 


0290 


0294 


107 


0294 


0298 


0302 


0306 0310 


0314 


0318 


0322 


0326 


0330 


0334 


108 


0334 


0338 


0342 


0346 0350 


0354 


0358 


0362 


0366 


0370 


0374 


109 
110 


0374 


0378 


0382 


0386 0390 


0394 


0398 


0402 


0406 


0410 


0414 


0414 


0418 


0422 


0426 0430 


0434 


0438 


0441 


0445 


0449 


0453 


111 


0453 


0457 


0461 


0465 0469 


0473 


0477 


0481 


0484 


0488 


0492 


112 


0492 


0496 


0500 


0504 0508 


0512 


0515 


0519 


0523 


0527 


0531 


113 


0531 


0535 


0538 


0542 0546 


0550 


0554 


0558 


0561 


0565 


0569 


114 
115 


0569 


0573 


0577 


0580 0584 


0588 


0592 


0596 


0599 


0603 


0607 


0607 


0611 


0615 


0618 0622 


0626 


0630 


0633 


0637 


0641 


0645 


116 


0645 


0648 


0652 


0656 0660 


0663 


0667 


0671 


0674 


0678 


0682 


117 


0682 


0686 


0689 


0693 0697 


0700 


0704 


0708 


0711 


0715 


0719 


118 


0719 


0722 


0726 


0730 0734 


0737 


0741 


0745 


0748 


0752 1 0755 


119 
120 


0755 


0759 


0763 


0766 0770 


0774 


0777 


0781 


0785 


0788 \ 0792 


0792 


0795 


0799 


0803 0806 


0810 


0813 


0817 


0821 


0824 


0828 


121 


0828 


0831 


0835 


0839 0842 ' 0846 


0849 


0853 


0856 


0860 


0864 


122 


0864 


0867 


0871 


0874 0878 / 0881 


0885 


0888 


0892 


0896 


0899 


123 


0899 


0903 


0906 


0910 0913 


0917 


0920 


0924 


0927 


0931 


0934 


124 
125 


0934 


0938 


0941 


0945 0948 


0952 


0955 


0959 


0962 


0966 


0969 


0969 


0973 


0976 


0980 0983 


0986 


0990 


0993 


0997 


1000 


1004 


126 


1004 


1007 


1011 


1014 1017 


1021 


1024 


1028 


1031 


1035 


1038 


127 


1038 


1041 


1045 


1048 1052 


1055 


1059 


1062 


1065 


1069 


1072 


128 


1072 


1075 


1079 


1082 1086 


1089 


1092 


1096 


1099 


1103 


1106 


129 
130 


1106 


1109 


1113 


1116 1119 


1123 


1126 


1129 


1133 


1136 


1139 


1139 


1143 


1146 


1149 1153 


1156 


1159 


1163 


1166 


1169 


1173 


131 


1173 


1176 


1179 


1183 1186 


1189 


1193 


1196 


1199 


1202 


1206 


132 


1208 


1209 


1212 


1216 1219 


1222 


1225 


1229 


1232 


1235 


1239 


133 


1239 


1242 


1245 


1248 1252 


1255 


1258 


1261 


1265 


1268 


1271 


134 
135 


1271 


1274 


1278 


1281 1284 


1287 


1290 


1294 


1297 


1300 


1303 


1303 


1307 


1310 


1313 1316 


1319 


1323 


1326 


1329 


1332 


1335 


136 


1335 


1339 


1342 


1345 1348 


1351 


1355 


1358 


1361 


1364 


1367 


137 


1367 


1370 


1374 


1377 1380 


1383 


1386 


1389 


1392 


1396 


1399 


138 


1399 


1402 


1405 


1408 1411 


1414 


1418 


1421 


1424 


1427 


1430 


139 
140 


1430 


1433 


1436 


1440 1443 


1446 


1449 


1452 


1455 


1458 


1461 


1461 


1464 


1467 


1471 1474 


1477 


1480 


1483 


1486 


1489 


1492 


141 


1492 


1495 


1498 


1501 1504 


1508 


1511 


1514 


1517 


1520 


1523 


142 


1523 


1526 


1529 


1532 1535 


1538 


1541 


1544 


1547 


1550 


1553 


143 


1553 


1556 


1559 


1562 1565 


1569 


1572 


1575 


1578 


1581 


1584 


144 
145 


1584 


1587 


1590 


1593 1596 


1599 


1602 


1605 


1608 


1611 


1614 


1614 


1617 


1620 


1623 1626 


1629 


1632 


1635 


1638 


1641 


1644 


146 


1644 


1647 


1649 


1652 1655 


1658 


1661 


1664 


1667 


1670 


1673 


147 


1673 


1676 


1679 


1682 1685 


1688 


1691 


1694 


1697 


1700 


1703 


148 


1703 


1706 


1708 


1711 1714 


1717 


1720 


1723 


1726 


1729 


1732 


149 


1732 


1735 


1738 


1741 1744 


1746 


1749 


1752 


1755 


1758 


1761 



TABLES. 

Logarithms. 



145 



N 





1 2 


3 


4 5 


6 


7 8 


9 


10 


150 


1761 


1764 1767 


1770 


1772 


1775 


1778 


1781 1784 


1787 


1790 


151 


1790 


1793 1796 


1798 


1801 


1804 


1807 


1810 1813 


1816 


1818 


152 


1818 


1821 1824 


1827 


1830 


1833 


1836 


1838 1841 


1844 


1847 


153 


1847 


1850 1853 


1855 


1858 


1861 


1864 


1867 1870 


1872 


1875 


154 
155 


1875 


1878 1881 


1884 


1886 


1889 


1892 


1895 1898 


1901 


1903 


1903 


1906 1909 


1912 


1915 


1917 


1920 


1923 1926 


1928 


1931 


156 


1931 


1934 1937 


1940 


1942 


1945 


1948 


1951 1953 


1956 


1959 


157 


1959 


1962 1965 


1967 


1970 


1973 


1976 


1978 1981 


1984 


1987 


158 


1987 


1989 1992 


1995 


1998 


2000 


2003 


2006 2009 


2011 


2014 


159 
160 


2014 


2017 2019 


2022 


2025 


2028 


2030 


2033 2036 


2038 


2041 


2041 


2044 2047 


2049 


2052 


2055 


2057 


2060 2063 


2066 


2068 


161 


2068 


2071 2074 


2076 


2079 


2082 


2084 


2087 2090 


2092 


2095 


162 


2095 


2098 2101 


2103 


2106 


2109 


2111 


2114 2117 


2119 


2122 


163 


2122 


2125 2127 


2130 


2133 


2135 


2138 


2140 2143 


2146 


2148 


164 
165 


2148 


2151 2154 


2156 


2159 


2162 


2164 


2167 2170 


2172 


2175 


2175 


2177 2180 


2183 


2185 


2188 


2391 


2193 2196 


2198 


2201 


166 


2201 


2204 2206 


2209 


2212 


2214 


2217 


2219 2222 


2225 


2227 


167 


2227 


2230 2232 


2235 


2238 


2240 


2243 


2245 2248 


2251 


2253 


168 


2253 


2256 2258 


2261 


2263 


2266 


2269 


2271 2274 


2276 


2279 


169 
170 


2279 


2281 2284 


2287 


2289 
2315 


2292 


2294 


2297 2299 


2302 


2304 
2330 


2304 


2307 2310 


2312 


2317 


2320 


2322 2325 


2327 


171 


2330 


2333 2335 


2338 


2340 


2343 


2345 


2348 2350 


2353 


2355 


172 


2355 


2358 2360 


2363 


2305 


2368 


2370 


2373 2375 


2378 


2380 


173 


2380 


2383 2385 


2388 


2390 


2393 


2395 


2398 2400 


2403 


2405 


174 
175 


2405 


2408 2410 


2413 


2415 


2418 


2420 


2423 2425 


2428 


2430 


2430 


2433 2435 


2438 


2440 


2443 


2445 


2448 2450 


2453 


2455 


176 


2455 


2458 2460 


2463 


2465 


2467 


2470 


2472 2475 


2477 


2480 


177 


2480 


2482 2485 


2487 


2490 


2492 


2494 


2497 2499 


2502 


2504 


178 


2504 


2507 2509 


2512 


2514 


2516 


2519 


2521 2524 


2526 


2529 


179 
180 


2529 


2531 2533 


2536 


2538 2541 


2543 


2545 2548 


2550 


2553 


2553 


2555 2558 


2560 


2562 ' 


2565 


2567 


2570 2572 


2574 


2577 


181 


2577 


2579 2582 


2584 


2586 


2589 


2591 


2594 2596 


2598 


2601 


182 


2601 


2603 2605 


2608 


2610 


2613 


2615 


2617 2620 


2622 


2625 


183 


2625 


2627 2629 


2632 


2634 


2636 


2639 


2641 2643 


2646 


2648 


184 
185 


2648 


2651 2653 2655 


2658 


2660 


2662 


2665 2667 


2669 


2672 


2672 


2674 2676 


2679 


2681 


2083 


2686 


2688 2690 


2693 


2695 


186 


2695 


2697 2700 


2702 


2704 


2707 


2709 


2711 2714 


2716 


2718 


187 


2718 


2721 2723 


2725 


2728 


2730 


2732 


2735 2737 


2739 


2742 


188 


2742 


2744 2746 


2749 


2751 


2753 


2755 


2758 2760 


2762 


2765 


189 
190 


2765 


2767 2769 


2772 


2774 


2776 


2778 


2781 2783 


2785 


2788 


2788 


2790 2792 


2794 


2797 


2799 


2801 


2804 2806 


2808 


2810 


191 


2810 


2813 2815 


2817 


2819 


2822 


2824 


2826 2828 


2831 


2833 


192 


2833 


2835 2838 


2840 


2842 


2844 


2847 


2849 2851 


2853 


2856 


193 


2856 


2858 2860 


2862 


2865 


2867 


2869 


2871 2874 


2876 


2878 


194 
195 


2878 


2880 2882 


2885 


2887 


2889 


2891 


2894 2896 


2898 


2900 


2900 


2903 2905 


2907 


2909 


2911 


2914 


2916 2918 


2920 


2923 


196 


2923 


2925 2927 


2929 


2931 


2934 


2936 


2938 2940 


2942 


2945 


197 


2945 


2947 2949 


2951 


2953 


2956 


2958 


2960 2962 


2964 


2967 


198 


2967 


2969 2971 


2973 


2975 


2978 


2980 


2982 2984 


2986 


2989 


199 


2989 


2991 2993 


2995 


2997 


2999 


3002 


3004 3006 


3008 


3010 



146 



TABLES. 



Trigonometric Functions. 



RADIANS. 


DEGREES. 


SINES. 


COSINES. 


TANGENTS. 


COTANGENTS. 










Nat. Log. 


Nat. Log. 


Nat. Log. 


Nat. 


Log. 






0.0000 


0°00' 


.0000 CO 


1.0000 0.0000 


.0000 00 


CO 


CO 


90° 00' 


1.5708 


0.0029 


10 


.0029 7.4637 


1.0000 .0000 


.0029 7.4637 


343.77 2.5363 


50 


1.5679 


0.0058 


20 


.0058 .7648 


1.0000 .0000 


.0058 .7648 


171.89 


.2352 


40 


1.5650 


0.0087 


30 


.0087 .9408 


1.0000 .0000 


.0087 .9409 


114.59 


.0591 


30 


1.5621 


0.0116 


40 


.0116 8.0658 


.9999 .0000 


.0116 8.0658 


85.940 


1.9342 


20 


1.5592 


0.0145 


50 


.0145 .1627 


.9999 .0000 


.0145 .1627 


68.750 


.8373 


10 


1.5563 


0.0175 


POO' 


.0175 8.2419 


.9998 9.9999 


.0175 8.2419 


57.290 


1.7581 


89° 00' 


1.5533 


0.02(H 


10 


.0204 .3088 


.9998 .9999 


.0204 .3089 


49.104 


.6911 


50 


1.5504 


0.0233 


20 


.0233 .3668 


.9997 .9999 


.0233 .3669 


42.964 


.6331 


40 


1.5475 


0.0262 


30 


.0262 .4179 


.9997 .9999 


.0262 .4181 


38.188 


.5819 


30 


1.5446 


0.0291 


40 


.0291 .4637 


.9996 .9998 


.0291 .4638 


34.368 


.5362 


20 


1.5417 


0.0320 


50 


.0320 .5050 


.9995 .9998 


.0320 .5053 


31.242 


.4947 


10 


1.5388 


0.0349 


2° 00' 


.0349 8.5428 


.9994 9.9997 


.0349 8.5431 


28.636 


1.4569 


88° 00' 


1.5359 


0.0378 


10 


.0378 .5776 


.9993 .9997 


.0378 .5779 


26.432 


.4221 


50 


1.5330 


0.0407 


20 


.0407 .6097 


.9992 .9996 


.0407 .6101 


24.542 


.3899 


40 


1.5301 


0.0436 


30 


.0436 .6397 


.9990 .9996 


.0437 .6401 


22.904 


.3599 


30 


1.5272 


0.0465 


40 


.0465 .6677 


.9989 .9995 


.0466 .6682 


21.470 


.3318 


20 


1.5243 


0.0495 


50 


.0494 .6940 


.9988 .9995 


.0495 .6945 


20.206 


.3055 


10 


1.5213 


0.0524 


3° 00' 


.0523 8.7188 


.9986 9.9994 


.0524 8.7194 


19.081 


1.2806 


87° 00' 


1.5184 


0.0553 


10 


.0552 .7423 


.9985 .9993 


.0553 .7429 


18.075 


.2571 


50 


1.5155 


0.0582 


20 


.0581 .7645 


.9983 .9993 


.0582 .7652 


17.169 


.2348 


40 


1.5126 


0.0611 


30 


.0610 .7857 


.9981 .9992 


.0612 .7865 


16.350 


.2135 


30 


1.5097 


0.0640 


40 


.0640 .8059 


.9980 .9991 


.0641 .8067 


15.605 


.1933 


20 


1.5068 


0.0669 


50 


.0669 .8251 


.9978 .9990 


.0670 .8261 


14.924 


.1739 


10 


1.5039 


0.0698 


4° 00' 


.0698 S.S436 


.9976 9.9989 


.0699 8.8446 


14.301 


1.1554 


86° 00' 


1.5010 


0.0727 


10 


.0727 .8613 


.9974 .9989 


.0729 .8624 


13.727 


.1376 


50 


1.4981 


0.0756 


20 


.0756 .8783 


.9971 .9988 


.0758 .8795 


13.197 


.1205 


40 


1.4952 


0.0785 


30 


.0785 .8946 


.9969 .9987 


.0787 .8960 


12.706 


.1040 


30 


1.4923 


0.0814 


40 


.0814 .9104 


.9967 .9986 


.0816 .9118 


12.251 


.0882 


20 


1.4893 


0.0844 


50 


.0843 .9256 


.9964 .9985 


.0846 .9272 


11.826 


.0728 


10 


1.4864 


0.0873 


5°00' 


.0872 8.9403 


.9962 9.9983 


.0875 8.9420 


11.430 


1.0580 


85° 00' 


1.4835 


0.0902 


10 


.0901 .9545 


.9959 .9982 


.0904 .9563 


11.059 


.0«7 


50 


1.4806 


0.0931 


20 


.0929 .9682 


.9957 .9981 


.0934 .9701 


]0.712 


.0299 


40 


1.4777 


0.0960 


30 


.0958 .9816 


.9954 .9980 


.0963 .9836 


10.385 


.0164 


30 


1.4748 


0.0989 


40 


.0987 .9945 


.9951 .9979 


.0992 .9966 


10.078 


.0034 


20 


1.4719 


0.1018 


50 


.1016 9.0070 


.9948 .9977 


.1022 9.0093 


9.7882 0.9907 


10 


1.4690 


0.1047 


6° 00' 


.1045 9.0192 


.9945 9.9976 


.1051 9.0216 


9.5144 


0.9784 


84° 00' 


1.4661 


0.1076 


10 


1074 .0311 


.9942 .9975 


.1080 .0336 


9.2553 


.9664 


50 


1.4632 


0.1105 


20 


.1103 .0426 


.9939 .9973 


.1110 .0453 


9.0098 


.9547 


40 


1.4603 


0.1134 


30 


.1132 .0539 


.9936 .9972 


.1139 .0567 


8.7769 


.9433 


30 


1.4574 


0.1164 


40 


.1161 .0648 


.9932 .9971 


.1169 .0678 


8.5555 


.9322 


20 


1.4544 


0.1193 


50 


.1190 .0755 


.9929 .9969 


.1198 .0786 


8.3450 


.9214 


10 


1.4515 


0.1222 


7° 00' 


.1219 9.0859 


.9925 9.9968 


.1228 9.0891 


8.1443 


0.9109 


83° 00' 


1.4486 


0.1251 


10 


.1248 .0961 


.9922 .9966 


.1257 .0995 


7.9530 


.9005 


50 


1.4457 


0.1280 


20 


.1276 .1060 


.9918 .9964 


.1287 .1096 


7.7704 


.8904 


40 


1.4428 


0.1309 


30 


.1305 .1157 


.9914 .9963 


.1317 .1194 


7.5958 


.8806 


30 


1.4399 


0.1338 


40 


.1334 .1252 


.9911 .9961 


.1346 .1291 


7.4287 


.8709 


20 


1.4370 


0.1367 


50 


.1363 .1345 


.9907 .9959 


.1376 .1385 


7.2687 


.8615 


10 


1.4341 


0.1396 


8° 00' 


.1392 9.1436 


.9903 9.9958 


.1405 9.1478 


7.1154 


0.8522 


82° 00' 


1.4312 


0.1425 


10 


.1421 .1525 


.9899 .9956 


.1435 .1569 


6.9682 


.8431 


50 


1.4283 


0.1454 


20 


.1449 .1612 


.9894 .9954 


.1465 .1658 


6.8269 


.8342 


40 


1.4254 


0.1484 


30 


.1478 .1697 


.9890 .9952 


.1495 .1745 


6.6912 


.8255 


30 


1.4224 


0.1513 


40 


.1507 .1781 


.9886 .9950 


.1524 .1831 


6.5606 


.8169 


20 


1.4195 


0.1542 


50 


.1536 .1863 


.9881 .9948 


.1554 .1915 


6.4348 


.8085 


10 


1.4166 


0.1571 


9° 00' 


.1564 9.1943 


.9877 9.9946 


.1584 9.1997 


6.3138 0.8003 


81° 00' 


1.4137 






Nat. Log. 


Nat. Log. 


Nat. Log. 


Nat. 


Log. 




■^ 






COSINES. 


SINES. 


COTANGENTS. 


TANGENTS. 


DEGREES. 


RADIANS. 



TABLES. 



147 



Trigonometric Functions. 



RADIANS. 


DEGREES. 


SINES. 


COSINES. 


TANGENTS. 


COTANGENTS. 










Nat. Log. 


Nat. Log. 


Nat. Log. 


Nat. Log. 






0.1571 


9° 00' 


.1564 9.1943 


.9877 9.9946 


.1584 9.1997 


6.3138 0.8003 


81° 00' 


1.4137 


0.1600 


10 


.1593 .2022 


.9872 .9944 


.1614 .2078 


6.1970 .7922 


50 


1.4108 


0.1629 


20 


.1622 .2100 


.9868 .9942 


.1644 .2158 


6.0S44 .7842 


40 


1.4079 


0.1658 


30 


.1650 .2176 


.9863 .9940 


.1673 .2236 


5.9758 .7764 


30 


1.4050 


0.1687 


40 


.1679 .2251 


.9858 .9938 


.1703 .2313 


5.8708 .7687 


20 


1.4021 


0.1716 


50 


.1708 .2324 


.9853 .9936 


.1733 .2389 


5.7694 .7611 


10 


1.3992 


0.1745 


10° 00' 


.1736 9.2397 


.9848 9.9934 


.1763 9.2463 


5.6713 0.7537 


80° 00' 


1.3963 


0.1774 


10 


.1765 .2468 


.9843 .9931 


.1793 .2536 


5.5764 .7464 


50 


1.3934 


0.1804 


20 


.1794 .2538 


.9838 .9929 


.1823 .2609 


5.4845 .7391 


40 


1.3904 


0.1833 


30 


.1822 .2606 


.9833 .9927 


.1853 .2680 


5.3955 .7320 


30 


1.3875 


0.1862 


40 


.1851 .2674 


.9827 .9924 


.1883 .2750 


5.3093 .7250 


20 


1.3846 


0.1891 


50 


.1880 .2740 


.9822 .9922 


.1914 .2819 


5.2257 .7181 


10 


1.3817 


0.1920 


11° 00' 


.1908 9.2806 


.9816 9.9919 


.1944 9.2S87 


5.1446 0.7113 


79° 00' 


1.3788 


0.1949 


10 


.1937 .2870 


.9811 .9917 


.1974 .2953 


5.0658 .7047 


50 


1.3759 


0.1978 


20 


.1965 .2934 


.9805 .9914 


.2004 .3020 


4.9894 .6980 


40 


1.3730 


0.2007 


30 


.1994 .2997 


.9799 .9912 


.2035 .3085 


4.9152 .6915 


30 


1.3701 


0.2036 


40 


.2022 .3058 


.9793 .9909 


.2065 .3149 


4.8430 .6851 


20 


1.3672 


0.2065 


50 


.2051 .3119 


.9787 .9907 


.2095 .3212 


4.7729 .6788 


10 


1.3643 


0.2094 


12° 00' 


.2079 9.3179 


.9781 9.9904 


.2126 9.3275 


4.7046 0.6725 


78° 00' 


1.3614 


0.2123 


10 


.2108 .3238 


.9775 .9901 


.2156 .3336 


4.6382 .6664 


50 


1.3584 


0.2153 


20 


.2136 .3296 


.9769 .9899 


.2186 .3397 


4.5736 .6603 


40 


1.3555 


0.2182 


30 


.2164 .3353 


.9763 .9896 


.2217 .3458 


4.5107 .6542 


30 


1.3526 


0.2211 


40 


.2193 .3410 


.9757 .9893 


.2247 .3517 


4.4494 .6483 


20 


1.3497 


0.2240 


50 


.2221 .3466 


.9750 .9890 


.2278 .3576 


4.3897 .6424 


10 


1.3468 


0.2269 


13° 00' 


.2250 9.3521 


.9744 9.9887 


.2309 9.3634 


4.3315 0.6366 


77° 00' 


1.3439 


0.2298 


10 


.2278 .3575 


.9737 .9884 


.2339 .3691 


4.2747 .6309 


50 


1.3410 


0.2327 


20 


.2306 .3629 


.9730 .9881 


.2370 .3748 


4.2193 .6252 


40 


1.3381 


0.2356 


30 


.2334 .3682 


.9724 .9878 


.2401 .3804 


4.1653 .6196 


30 


1.3352 


0.2385 


40 


.2363 .3734 


.9717 .9875 


.2432 .3859 


4.1126 .6141 


20 


1.3323 


0.2414 


50 


.2391 .3786 


.9710 .9872 


.2462 .3914 


4.0611 .6086 


10 


1.3294 


0.2443 


14° 00' 


.2419 9.3837 


.9703 9.9869 


.2493 9.3968 


4.0108 0.6032 


76° 00' 


1.3265 


0.2473 


10 


.2447 .3887 


.9696 .9866 


.2524 .4021 


3.9617 .5979 


50 


1.3235 


0.2502 


20 


.2476 .3937 


.9689 .9863 


.2555 .4074 


3.9136 .5926 


40 


1.3206 


0.2531 


30 


.2504 .3986 


.9681 .9859 


.2586 .4127 


3.8667 .5873 


30 


1.3177 


0.2560 


40 


.2532 .4035 


.9674 .9856 


.2617 .4178 


3.8208 .5822 


20 


1.3148 


0.2589 


50 


.2560 .4083 


.9667 .9853 


.2648 .4230 


3.7760 .5770 


10 


1.3119 


0.2618 


15° 00' 


.2588 9.4130 


.9659 9.9849 


.2679 9.4281 


3.7321 0.5719 


75° 00' 


1.3090 


0.2647 


10 


.2616 .4177 


.9652 .9846 


.2711 .4331 


3.6891 .5669 


50 


1.3061 


0.2676 


20 


.2644 .4223 


.9644 .9843 


.2742 .4381 


3.6470 .5619 


40 


1.3032 


0.2705 


30 


.2672 .4269 


.9636 .9839 


.2773 .4430 


3.6059 .5570 


30 


1.3003 


0.2734 


40 


.2700 .4314 


.9628 .9836 


.2805 .4479 


3.5656 .5521 


20 


1.2974 


0.2763 


50 


.2728 .4359 


.9621 .9832 


.2836 .4527 


3.5261 .5473 


10 


1.2945 


0.2793 


16° 00' 


.2756 9.4403 


.9613 9.9828 


.2867 9.4575 


3.4874 0.5425 


74° 00' 


1.2915 


0.2822 


10 


.2784 .4447 


.9605 .9825 


.2899 .4622 


3.4495 .5378 


50 


1.2886 


0.2851 


20 


.2812 .4491 


.9596 .9821 


.2931 .4669 


3.4124 .5331 


40 


1.2857 


0.2880 


30 


.2840 .4533 


.9588 .9817 


.2962 .4716 


3.3759 .5284 


30 


1.2828 


0.2909 


40 


.2868 .4576 


.9580 .9814 


.2994 .4762 


3.3402 .5238 


20 


1.2799 


0.2938 


50 


.2896 .4618 


.9572 .9810 


.3026 .4808 


3.3052 .5192 


10 


1.2770 


0.2967 


17° 00' 


.2924 9.4659 


.9563 9.9806 


.3057 9.4853 


3.2709 0.5147 


73° 00' 


1.2741 


0.2996 


10 


.2952 .4700 


.9555 .9802 


.3089 .4898 


3.2371 .5102 


50 


1.2712 


0.3025 


20 


.2979 .4741 


.9546 .9798 


.3121 .4943 


3.2041 .5057 


40 


1.2683 


0.3054 


30 


.3007 .4781 


.9537 .9794 


.3153 .4987 


3.1716 .5013 


30 


1.2654 


0.3083 


40 


.3035 .4821 


.9528 .9790 


.3185 .5031 


3.1397 .4969 


20 


1.2625 


0.3113 


50 


.3062 .4861 


.9520 .9786 


.3217 .5075 


3.1084 .4925 


10 


1.2595 


0.3142 


18° 00' 


.3090 9.4900 


.9511 9.9782 


.3249 9.5118 


3.0777 0.4882 


72° 00' 


1.2566 






Nat. Log. 


Nat. Log. 


Nat. Log. 


Nat. Log. 










COSINES. 


SINES. 


COTANGENTS. 


TANGENTS. 


DEGREES. 


RADIANS. 



148 



TABLES. 



Trigonometric Functions. 



RADIANS. 


DEGREES. 


SINES. 


COSINES. 


TANGENTS. 


COTANGENTS. 










Nat. Log. 


Nat. Log. 


Nat. Log. 


i Nat. 


Log. 






0.3142 


18° 00' 


.3090 9.4900 


.9511 9.9782 


.3249 9.5118 


3.0777 0.4882 


72° 00' 


1.2566 


0.3171 


10 


.3118 .4939 


.9502 .9778 


.3281 .5161 


3.0475 


.4839 


50 


1.2537 


0.3200 


20 


.3145 .4977 


.9492 .9774 


.3314 .5203 


3.0178 


.4797 


40 


1.2508 


0.3229 


30 


.3173 .5015 


.9483 .9770 


.3346 .5245 


' 2.9887 


.4755 


30 


1.2479 


0.3258 


40 


.3201 .5052 


.9474 .9765 


.3378 .5287 


2.9600 


.4713 


20 


1.2450 


0.3287 


50 


.3228 .5090 


.9465 .9761 


.3411 .5329 


2.9319 


.4671 


10 


1.2421 


0.3316 


19° 00' 


.3256 9.5126 


.9455 9.9757 


.3443 9.5370 


2.9042 


0.4630 


71° 00' 


1.2392 


0.3345 


10 


.3283 .5163 


.9446 .9752 


.3476 .5411 


2.8770 


.4589 


50 


1.2363 


0.3374 


20 


.3311 .5199 


.9436 .9748 


.3508 .5451 


2.8502 


.4549 


40 


1.2334 


0.3403 


30 


.3338 .5235 


.9426 .9743 


.3541 .5491 


2.8239 


.4509 


30 


1.2305 


0.3432 


40 


.3365 .5270 


.9417 .9739 


.3574 .5531 


2.7980 


.4469 


20 


1.2275 


0.3462 


50 


.3393 .5306 


.9407 .9734 


.3607 .5571 


2.7725 


.4429 


10 


1.2246 


0.3491 


20° 00' 


.3420 9.5341 


.9397 9.9730 


.3640 9.5611 


2.7475 


0.4389 


70° 00' 


1.2217 


0.3520 


10 


.3448 .5375 


.9387 .9725 


.3673 .5650 


2.7228 


.4350 


SO 


1.2188 


0.3549 


20 


.3475 .5409 


.9377 .9721 


.3706 .5689 


2.6985 


.4311 


40 


1.2159 


0.3578 


30 


.3502 .5443 


.9367 .9716 


.3739 .5727 


2.6746 


.4273 


30 


1.2130 


0.3607 


40 


.3529 .5477 


.9356 .9711 


.3772 .5766 


2.6511 


.4234 


20 


1.2101 


0.3636 


50 


.3557 .5510 


.9346 .9706 


.3805 .5804 


2.6279 


.4196 


10 


1.2072 


0.3665 


21° 00' 


.3584 9.5543 


.9336 9.9702 


.3839 9.5842 


2 6051 


0.4158 


69° 00' 


1.2043 


0.3694 


10 


.3611 .5576 


.9325 .9697 


.3872 .5879 


2.5826 


.4121 


50 


1.2014 


0.3723 


20 


.3638 .5609 


.9315 .9692 


.3906 .5917 


2.5605 


.4083 


40 


1.1985 


0.3752 


30 


.3665 .5641 


.9304 .9687 


.3939 .5954 


25386 


.4046 


30 


1.1956 


0.3782 


40 


.3692 .5673 


.9293 .9682 


.3973 .5991 


2.5172 


.4009 


20 


1.1926 


0.3811 


50 


.3719 .5704 


.9283 .9677 


.4006 .6028 


2.4960 


.3972 


10 


1.1897 


0.3840 


22° 00' 


.3746 9.5736 


.9272 9.9672 


.4040 9.6064 


2.4751 


0.3936 


68° 00' 


1.1868 


0.3869 


10 


.3773 .5767 


.9261 .9667 


.4074 .6100 


2.4545 


.3900 


50 


1.1839 


0.3898 


20 


.3800 .5798 


.9250 .9661 


.4108 .6136 


2.4342 


.3864 


40 


1.1810 


0.3927 


30 


.3827 .5828 


.9239 .9656 


.4142 .6172 


2.4142 


.3828 


30 


1.1781 


0.3956 


40 


.3854 .5859 


.9228 .9651 


.4176 .6208 


2.3945 


.3792 


20 


1.1752 


0.3985 


SO 


.3831 .5889 


.9216 .9646 


.4210 .6243 


23750 


.3757 


10 


1.1723 


0.4014 


23° 00' 


.3907 9.S919 


.9205 9.9640 


.4245 9.6279 


2.3559 0.3721 


67° 00' 


1.1694 


0.4043 


10 


.3934 .5948 


.9194 .9635 


.4279 .6314 


2.3369 


.3686 


SO 


1.1665 


0.4072 


20 


.3961 .5978 


.9182 .9629 


.4314 .6348 


2.3183 


.3652 


40 


1.1636 


0.4102 


30 


.3987 .6007 


.9171 .9624 


.4348 .6383 


2.2998 


.3617 


30 


1.1606 


0.4131 


40 


.4014 , .6036 


.9159 .9618 


.4383 .6417 


2.2817 


.3583 


20 


1.1577 


0.4160 


50 


.4041 .6065 


.9147 .9613 


.4417 .6452 


2.2637 


.3548 


10 


1.1548 


0.4189 


24° 00' 


.4067 9.6093 


.9135 9.9607 


.4452 9.6486 


2.2460 0.3514 


66° 00' 


1.1519 


0.4218 


10 


.4094 .6121 


.9124 .9602 


.4487 .6520 


2.2286 


.3480 


SO 


1.1490 


0.4247 


20 


.4120 .6149 


.9112 .9596 


.4522 .6553 


2.2113 


.3447 


40 


1.1461 


0.4276 


30 


.4147 .6177 


.9100 .9590 


.4557 .6587 


2.1943 


.3413 


30 


1.1432 


0.4305 


40 


.4173 .6205 


.9088 .9584 


.4592 .6620 


2.1775 


.3380 


20 


1.1403 


0.4334 


50 


.4200 .6232 


.9075 .9579 


.4628 .6654 


2.1609 


.3346 


10 


1.1374 


0.4363 


25° 00' 


.4226 9.6259 


.9063 9.9573 


.4663 9.6687 


2 1445 


0.3313 


65° 00' 


1.1345 


0.4392 


10 


.4253 .6286 


.9051 .9567 


.4699 .6720 


2.1283 


.3280 


50 


1.1316 


0.4422 


20 


.4279 .6313 


.9038 .9561 


.4734 .6752 


2.1123 


.3248 


40 


1.1286 


0.4451 


30 


.4305 .6340 


.9026 .9555 


.4770 .6785 


2.0965 


.3215 


30 


1.1257 


0.4480 


40 


.4331 .6366 


.9013 .9549 


.4806 .6817 


2.0809 


.3183 


20 


1.1228 


0.4509 


50 


.4358 .6392 


.9001 .9543 


.4841 .6850 


2.0655 


.3150 


10 


1.1199 


0.4538 


26° 00' 


.4384 9.6418 


.8988 9.9537 


.4877 9.6882 


2.0503 0.3118 


64° 00' 


1.1170 


0.4567 


10 


.4410 .6444 


.8975 .9530 


.4913 .6914 


2.0353 


.3086 


50 


1.1141 


0.4596 


20 


.4436 .6470 


.8962 .9524 


.4950 .6946 


2.0204 


.3054 


40 


1.1112 


0.4625 


30 


.4462 .6495 


.8949 .9518 


.4986 .6977 


2.0057 


.3023 


30 


1.1083 


0.4654 


40 


.4488 .6521 


.8936 .9512 


.5022 .7009 


1.9912 


.2991 


20 


1.1054 


0.4683 


SO 


.4514 .6546 


.8923 .9505 


.5059 .7040 


1.9768 


.2960 


10 


1.1025 


0.4712 


27° 00' 


.4540 9.6570 


.8910 9.9499 


.5095 9.7072 


1.9626 0.2928 


63° 00' 


1.0996 






Nat. Log. 


Nat. Log. 


Nat. Log. 


Nat. 


Log. 










COSINES. 


SINES. 


COTANGENTS. 


TANGENTS. 


DEGREES. 


RADIANS. 



TABLES. 



149 









Trigonometric Functions. 








RADIANS. 


DEGREES. 


SINES. 


COSINES. 


TANGENTS. 


COTANGENTS. 










Nat. Log. 


Nat. Log. 


Nat. 


Log. 


Nat. Log. 






0.4712 


27° 00' 


.4540 9.6570 


.8910 9.9499 


.5095 


9.7072 


1.9626 0.2928 


63° 00' 


1.0996 


0.4741 


10 


.4566 .6595 


.8897 .9492 


.5132 


.7103 


1.9486 .2897 


50 


1.0966 


0.4771 


20 


.4592 .6620 


.8884 .9486 


.5169 


.7134 


1.9347 .2866 


40 


1.0937 


0.4800 


30 


.4617 .6644 


.8870 .9479 


.5206 


,7165 


1.9210 .2835 


30 


1.0908 


0.4829 


40 


.4643 .6668 


.8857 .9473 


.5243 


.7196 


1.9074 .2804 


20 


1.0879 


0.4858 


50 


.4669 .6692 


.8843 .9466 


.5280 


.7226 


1.8940 .2774 


10 


1.0850 


0.4887 


28° 00' 


.4695 9.6716 


.8829 9.9459 


.5317 9.7257 


1.8807 0.2743 


62° 00' 


1.0821 


0.4916 


10 


.4720 .6740 


.8816 .9453 


.5354 


.7287 


1.8676 .2713 


50 


1.0792 


0.4945 


20 


.4746 .6763 


.8802 .9446 


.5392 


.7317 


1.8546 .2683 


40 


1.0763 


0.4974 


30 


.4772 .6787 


.8788 .9439 


.5430 


.7348 


1.8418 .2652 


30 


1.0734 


0.5003 


40 


.4797 .6810 


.8774 .9432 


.5467 


.7378 


1.8291 .2622 


20 


1.0705 


0.5032 


50 


.4823 .6833 


.8760 .9425 


.5505 


.7408 


1.S165 .2592 


10 


1.0676 


0.5061 


29° 00' 


.4848 9.6856 


.8746 9.9418 


.5543 9.7438 


1.8040 0.2562 


61° 00' 


1.0647 


0.5091 


10 


.4874 .6878 


.8732 .9411 


.5581 


.7467 


1.7917 .2533 


50 


1.0617 


0.5120 


20 


.4899 .6901 


.8718 .9404 


.5619 


.7497 


1.7796 .2503 


40 


1.0588 


0.5149 


30 


.4924 .6923 


.8704 .9397 


.5658 


.7526 


1.7675 .2474 


30 


1.0559 


0.5178 


40 


.4950 .6946 


.8689 .9390 


.5696 


.7556 


1.7556 .2444 


20 


1.0530 


0.5207 


50 


.4975 .6968 


.8675 .9383 


.5735 


.7585 


1.7437 .2415 


10 


1.0501 


0.5236 


30° 00' 


.5000 9.6990 


.8660 9.9375 


.5774 9.7614 


1.7321 0.2386 


60° 00' 


1.0472 


0.5265 


10 


.5025 .7012 


.8646 .9368 


.5812 


.7644 


1.7205 .2356 


50 


1.0443 


0.5294 


20 


.5050 .7033 


.8631 .9361 


.5851 


.7673 


1.7090 .2327 


40 


1.0414 


0.5323 


30 


.5075 .7055 


.8616 .9353 


.5890 


.7701 


1.6977 .2299 


30 


1.0385 


0.5352 


40 


.5100 .7076 


.8601 .9346 


.5930 


.7730 


1.6864 .2270 


20 


1.0356 


0.5381 


50 


.5125 .7097 


.8587 .9338 


.5969 


.7759 


1.6753 .2241 


10 


1.0327 


0.5411 


31° 00' 


.5150 9.7118 


.8572 9.9331 


.6009 9.7788 


1.6643 0.2212 


59° 00' 


1.0297 


0.5440 


10 


.5175 .7139 


.8557 .9323 


.6048 


.7816 


1.6534 .2184 


50 


1.0268 


0.5469 


20 


.5200 .7160 


.8542 .9315 


.6088 


.7845 


1.6426 .2155 


40 


1.0239 


0.5498 


30 


.5225 .7181 


.8526 .9.308 


.6128 


.7873 


1.6319 .2127 


30 


1.0210 


0.5527 


40 


.5250 .7201 


.8511 .9300 


.6168 


.7902 


1.6212 .2098 


20 


1.0181 


0.5556 


50 


.5275 .7222 


.8496 .9292 


.6208 


.7930 


1.6107 .2070 


10 


1.0152 


0.5585 


32° 00' 


.5299 9.7242 


.8480 9.9284 


.6249 9.7958 


1.6003 0.2042 


58° 00' 


1.0123 


0.5614 


10 


.5324 .7262 


.8465 .9276 


.6289 


.7986 


1.5900 .2014 


50 


1.0094 


0.5643 


20 


.5348 .7282 


.8450 .9268 


.6330 


.8014 


1.5798 .1986 


40 


1.0065 


0.5672 


30 


.5373 .7302 


.8434 .9260 


.6371 


.8042 


1.5697 .1958 


30 


1.0036 


0.5701 


40 


.5398 .7322 


.8418 .9252 


.6412 


.8070 


1.5597 .1930 


20 


1.0007 


0.5730 


50 


.5422 .7342 


.8403 .9244 


.6453 


.8097 


1.5497 .1903 


10 


0.9977 


0.5760 


33° 00' 


.5446 9.7361 


.8387 9.9236 


.6494 9.8125 


1.5399 0.1875 


57° 00' 


0.9948 


0.5789 


10 


.5471 .7380 


.8371 .9228 


.6536 


.8153 


1.5301 .1847 


50 


0.9919 


0.5818 


20 


.5495 .7400 


.8355 .9219 


.6577 


.8180 


1.5204 .1820 


40 


0.9890 


0.5847 


30 


.5519 .7419 


.8339 .9211 


.6619 


.8208 


1.5108 .1792 


30 


0.9861 


0.5876 


40 


.5544 .7438 


.8323 .9203 


.6661 


.8235 


1.5013 .1765 


20 


0.9832 


0.5905 


50 


.5568 .7457 


.8307 .9194 


.6703 


.8263 


1.4919 .1737 


10 


0.9803 


0.5934 


34° 00' 


.5592 9.7476 


.8290 9.9186 


.6745 


9.8290 


1.4826 0.1710 


56° 00' 


0.9774 


0.5963 


10 


.5616 .7494 


.8274 .9177 


.6787 


.8317 


1.4733 .1683 


50 


0.9745 


0.5992 


20 


.5640 .7513 


.8258 .9169 


.6830 


.8344 


1.4641 .1656 


40 


0.9716 


0.6021 


30 


.5664 .7531 


.8241 .9160 


.6873 


.8371 


1.4550 .1629 


30 


0.9687 


0.6050 


40 


.5688 .7550 


.8225 .9151 


.6916 


.8398 


1.4460 .1602 


20 


0.9657 


0.6080 


50 


.5712 .7568 


.8208 .9142 


.6959 


.8425 


1.4370 .1575 


10 


0.9628 


0.6109 


35° 00' 


.5736 9.7586 


.8192 9.9134 


.7002 


9.8452 


1.4281 0.1548 


55° 00' 


0.9599 


0.6138 


10 


.5760 .7604 


.8175 .9125 


.7046 


.8479 


1.4193 .1521 


50 


0.9570 


0.6167 


20 


.5783 .7622 


.8158 .9116 


.7089 


.8506 


1.4106 .1494 


40 


0.9541 


0.6196 


30 


.5807 .7640 


.8141 .9107 


.7133 


.8533 


1.4019 .1467 


30 


0.9512 


0.6225 


40 


.5831 .7657 


.8124 .9098 


.7177 


.8559 


1.3934 .1441 


20 


0.9483 


0.6254 


50 


.5854 7675 


.8107 .9089 


.7221 


.8586 


1.3848 .1414 


10 


0.9454 


0.6283 


36° 00' 


.5878 9.7692 


.8090 9.9080 


.7265 


9.8613 


1.3764 0.1387 


54° 00' 


0.9425 






Nat. Log. 


Nat. Log. 


Nat. 


Log. 


Nat. Log. 










COSINES. 


SINES. 


COTANGENTS. 


TANGENTS. 


DEGREES. 


RADIANS. 



150 






TABLES. 














Trigonometric Functions. 




RADIANS. 


DEGREES. 


SINES. 


COSINES. 


TANGENTS. 


COTANGENTS. 










Nat. Log. 


Nat. Log. 


Nat. Log. 


Nat. Log. 






0.62S3 


36° 00' 


.5878 9.7692 


.8090 9.9080 


.7265 9.8613 


1.3764 0.13S7 


54° 00' 


0.9425 


0.6312 


10 


.5901 .7710 


.8073 .9070 


.7310 .8639 


1.3680 .1361 


50 


0.9396 


0.6341 


20 


.5925 .7727 


.8056 .9061 


.7355 .8666 


1.3597 .1334 


40 


0.9367 


0.6370 


30 


.5948 .7744 


.8039 .9052 


.7400 .8692 


1.3514 .1308 


30 


0.9338 


0.6400 


40 


.5972 .7761 


.8021 .9042 


.7445 .8718 


1.3432 .1282 


20 


0.9308 


0.6429 


50 


.5995 .7778 


.8004 .9033 


.7490 .8745 


1.3351 .1255 


10 


0.9279 


0.6458 


37° 00' 


.6018 9.7795 


.7986 9.9023 


.7536 9.8771 


1.3270 0.1229 


53° 00' 


0.9250 


0.64S7 


10 


.6041 .7811 


.7969 .9014 


.7581 .8797 


1.3190 .1203 


50 


0.9221 


0.6516 


20 


.6065 .7828 


.7951 .9004 


.7627 .8824 


1.3111 .1176 


40 


0.9192 


0.6545 


30 


.6088 .7844 


.7934 .8995 


.7673 .8850 


1.3032 .1150 


30 


0.9163 


0.6574 


40 


.6111 .7861 


.7916 .8985 


.7720 .8876 


1.2954 .1124 


20 


0.9134 


0.6603 


50 


.6134 .7877 


.7898 .8975 


.7766 .8902 


1.2876 .1098 


10 


0.9105 


0.6632 


38° 00' 


.6157 9.7893 


.7880 9.8965 


.7813 9.8928 


1.2799 0.1072 


52° 00' 


0.9076 


0.6661 


10 


.6180 .7910 


.7862 .8955 


.7860 .8954 


1.2723 .1046 


50 


0.9047 


0.6690 


20 


.6202 .7926 


.7844 .8945 


.7907 .8980 


1.2647 .1020 


40 


0.9018 


0.6720 


30 


.6225 .7941 


.7826 .8935 


.7954 .9006 


1.2572 .0994 


30 


0.8988 


0.6749 


40 


.6248 .7957 


.7808 .8925 


.8002 .9032 


1.2497 .0968 


20 


0.8959 


0.67 7S 


50 


.6271 .7973 


.7790 .8915 


.8050 .9058 


1.2423 .0942 


10 


0.8930 


0.6807 


39° 00' 


.6293 9.7989 


.7771 9.8905 


.8098 9.9084 


1.2349 0.0916 


51° 00' 


0.8901 


0.6836 


10 


.6316 .8004 


.7753 .8895 


.8146 .9110 


1.2276 .0890 


50 


0.8872 


0.6865 


20 


.6338 .8020 


.7735 .8884 


.8195 .9135 


1.2203 .0865 


40 


0.8843 


0.6894 


30 


.6361 .8035 


.7716 .8874 


.8243 .9161 


1.2131 .0839 


30 


0.8814 


0.6923 


40 


.6383 .8050 


.7698 .8864 


.8292 .9187 


1.2059 .0813 


20 


0.8785 


0.6952 


50 


.6406 .8066 


.7679 .8853 


.8342 .9212 


1.1988 .0788 


10 


0.8756 


0.6981 


40° 00' 


.6428 9.S081 


.7660 9.8843 


.8391 9.9238 


1.1918 0.0762 


50° 00' 


0.8727 


0.7010 


10 


.6450 .8096 


.7642 .8832 


.8441 .9264 


1.1847 .0736 


50 


0.S698 


0.7039 


20 


.6472 .8111 


.7623 .8821 


.8491 .9289 


1.1778 .0711 


40 


0.8668 


0.7069 


30 


.6494 .8125 


.7604 .8810 


.8541 .9315 


1.1708 .0685 


30 


0.8639 


0.7098 


40 


.65.'.7 .8140 


.7585 .8800 


.8591 .9341 


1.1640 .0659 


20 


0.S610 


0.7127 


50 


.6539 .8155 


.7566 .8789 


.8642 .9366 


1.1571 .0634 


10 


0.8581 


0.7156 


41° 00' 


.6561 9.8169 


.7547 9.8778 


.8693 9.9392 


1.1504 0.0608 


49° 00' 


0.8552 


0.7185 


10 


.6583 .8184 


.7528 .8767 


.8744 .9417 


1.1436 .0583 


50 


0.8523 


0.7214 


20 


.6604 .8198 


.7509 .8756 


.8796 .9443 


1.1369 .0557 


40 


0.8494 


0.7243 


30 


.6626 .8213 


.7490 .8745 


.8847 .9468 


1.1303 .0532 


30 


0.8465 


0.7272 


40 


.6648 .8227 


.7470 .8733 


.8899 .9494 


1.1237 .0506 


20 


0.8436 


0.7301 


50 


.6670 .8241 


.7451 .8722 


.8952 .9519 


1.1171 .0481 


10 


0.8407 


0.7330 


42° 00' 


.6691 9.8255 


.7431 9.8711 


.9004 9.9544 


1.1106 0.0456 


48° 00' 


0.8378 


0.7359 


10 


.6713 .8269 


.7412 .8699 


.9057 .9570 


1.1041 .0430 


50 


0.8348 


0.7389 


20 


.6734 .8283 


.7392 .8688 


.9110 .9595 


1.0977 .0405 


40 


0.8319 


0.7418 


30 


.6756 .8297 


.7373 .8676 


.9163 .9621 


1.0913 .0379 


30 


0.8290 


0.7447 


40 


.6777 .8311 


.7353 .8665 


.9217 .9646 


1.0850 .0354 


20 


0.8261 


0.7476 


50 


.6799 .8324 


.7333 .8653 


.9271 .9671 


1.0786 .0329 


10 


0.8232 


0.7505 


43° 00' 


.6820 9.8338 


.7314 9.8641 


.9325 9.9697 


1.0724 0.0303 


47° 00' 


0.8203 


0.7534 


10 


.6841 .8351 


.7294 .8629 


.9380 .9722 


1.0661 .0278 


50 


0.8174 


0.7563 


20 


.6862 .8365 


.7274 .8618 


.9435 .9747 


1.0599 .0253 


40 


0.8145 


0.7592 


30 


.6884 .8378 


.7254 .8606 


.9490 .9772 


1.0538 .0228 


30 


0.8116 


0.7621 


40 


.6905 .8391 


.7234 .8594 


.9545 .9798 


1.0477 .0202 


20 


0.8087 


0.7650 


50 


.6926 .8405 


.7214 .8582 


.9601 .9823 


l.ail6 .0177 


10 


0.8058 


0.7679 


44° 00' 


.6947 9.8418 


.7193 9.8569 


.9657 9.9848 


1.0355 0.0152 


46° 00' 


0.8029 


0.7709 


10 


.6967 .8431 


.7173 .8557 


.9713 .9874 


1.0295 .0126 


50 


0.7999 


0.7738 


20 


.6988 .8444 


.7153 .8545 


.9770 .9899 


1.0235 .0101 


40 


0.7970 


0.7767 


30 


.7009 .8457 


.7133 .8532 


.9827 .9924 


1.0176 .0076 


30 


0.7941 


0.7796 


40 


.7030 .8469 


.7112 .8520 


.9884 .9949 


1.0117 .0051 


20 


0.7912 


0.7825 


50 


.7050 .8482 


.7092 .8507 


.9942 .9975 


1.0058 .0025 


10 


0.7883 


0.7854 


45° 00' 


.7071 9.8495 


.7071 9.8495 


1.0000 0.0000 


1.0000 0.0000 


45° 00' 


0.7854 






Nat. Log. 


Nat. Log. 


Nat. Log. 


Nat. Log. 










COSINES. 


SINES. 


COTANGENT.S. 


TANGENTS. 


DEGREES. 


RADIANS. 



TABLES. 



151 



Equivalents of Radians in Degrees, Minutes, and Seconds of Arc. 



RAmATJS- 


EQUIVALENTS. 


RADIANS. 


EQUIVALENTS. 


0.0001 


0° 0' 20".6 or 0°.005730 


0.0600 


3° 26' 15".9 


or 3°.437747 


0.0002 


0° 0'41".3 or 0°.011459 


0.0700 


4° 0'3S".5 


or 4°.010705 


0.0003 


0° 1'01".9 or 0°.017189 


O.OSOO 


4°35'01".2 


or 4°.5S3662 


0.0004 


0° V 22".5 or 0°. 022918 


0.0900 


5° 9'23".S 


or 5°.156620 


0.0005 


0° 1' 43". 1 or 0°.028648 


0.1000 


5° 43' 46". 5 


or 5°.729578 


0.0006 


0° 2'03".8 or 0°.034377 


0.2000 


11°27'33".0 


or 11°.459156 


0.0007 


Qo 2'24".4 or 0°.040107 


0.3000 


17°11'19".4 


or 17°. 188734 


0.0008 


0° 2'45".0 or 0°.045837 


0.4000 


22°55'05".9 


or 22°.918312 


0.0009 


0° 3'05".6 or 0°.051566 


0.5000 


2S°38'52".4 


or 28°.647890 


0.0010 


0° 3'26".3 or 0°. 057296 


0.6000 


34° 22' 3S".9 


or 34°377468 


0.0020 


0° 6'52".5 or 0°.114S92 


0.7000 


40° 6'25".4 


or 40°. 107046 


0.0030 


0°10'1S".8 or 0°.171887 


O.SOOO 


45°50'11".8 


or 45°.836624 


0.0040 


0°13'45".l or 0°. 229183 


0.9000 


51° 33' 58".3 


or 51°.566202 


0.0050 


0°17'11".3 or 0°.286479 


1.0000 


57°17'44".8 


or 57°.295780 


0.0060 


0°20'37".6 or 0^.343775 


2.0000 


114° 35' 29".6 


or 1M°.591559 


0.0070 


0°24'03".9 or 0°.401070 


3.0000 


171° 53' 14".4 


or 171°.8S7339 


0.00S0 


0°27'30".l or 0°. 458366 


4.0000 


229° 10' 59".2 


or 229°. 183 118 


0.0090 


0°30'S6".4 or 0°.515662 


5.0000 


286°28'44".0 


or 286°.478898 


0.0100 


0°34'22".6 or 0°.572958 


6.0000 


343°46'28".8 


or 343°.774677 


0.0200 


1° 8'45".3 or P.145916 


7.0000 


401° 4' 13" 6 


or 401°.070457 


0.0300 


1''43'07".9 or 10.718873 


8.0000 


458° 21' 58".4 


or 458°.366236 


0.0400 


2°17'30".6 or 2°.291831 


9.0000 


515°39'43".3 


or 515°.662016 


0.0500 


2° 51' S3".2 or 2°.864789 


10.0000 


572° 57' 2S".l 


or 572°.95779S 



The Values in Circular Measure of Angles which are given in 
Degrees and Minutes. 



r 


0.0003 


9' 


0.0026 


3° 


0.0524 


20° 


0.3491 


100° 


\.7453 


2' 


0.0006 


10' 


0.0029 


4° 


0.0698 


30° 


0.5236 


110° 


1.9199 


3' 


0.0009 


20' 


0.0058 


5° 


0.0873 


40° 


0.6981 


120° 


2.0944 


4' 


0.0012 


30' 


0.0087 


6° 


0.1047 


50° 


0.8727 


130° 


2.2689 


5' 


0.0015 


40' 


0.0116 


7° 


0.1222 


60° 


1.0472 


140° 


2.4435 


6' 


0.0017 


50' 


0.0145 


8° 


0.1396 


70° 


1.2217 


150° 


2.6180 


7' 


0.0020 


1° 


0.0175 


9° 


0.1571 


80° 


1.3963 


160° 


2.7925 


8' 


0.0023 


2° 


0.0349 


10° 


0.1745 


90° 


1.5708 


170° 


2.9671 



PAGE INDEX. 



INTEGRALS. 

Fundamental forms 

Rational algebraic expressions involving (a + bx) and (a' + b'x) 

{a + bx") . 
" " " (a + bx + cx^) . 

(a' + 6'x)and(a + 6a; + cx2) 
Rational fractions ......... 

Irrational algebraic expressions involving Va + bx or \/a + bx . 



(1 






11 



Miscellaneous algebraic expressions 

General transcendental forms ...... 

Expressions involving simple direct trigonometric functions 
Expressions involving inverse trigonometric functions 
Exponential forms ..... 

Logarithmic forms ..... 

Expressions involving hyperbolic functions 

Miscellaneous definite integrals 

Elliptic integrals 



Pages 

3,4 

5,7 

8,9 

10,11 

11-13 

13,14 

16,17 

18,19 

20-23 

31 

23-27 

(a' + b'x) and Va + bx + cx^ 27-30 

32-34 
35-37 
38-51 
51-53 
63-56 
66-58 
68-61 
62-65 
66-72 



V a + bx and Va' + b' x 
•V x2 ± a2 or Va2 — x^ 

V2 



ax — x^ 



Va + bx + cx2 



AUXILIARY FORMULAS AND TABLES. 



Trigonometric functions . 

Hyperbolic functions 

Elliptic functions, Bessel's functions 

Series 

Derivatives .... 

Green's Theorem and allied formulas 

Table of mathematical constants 

General formulas of integration 

Note on interpolation 

Table of the probability integral 

Tables of elliptic integrals 

Table of hyperbolic functions . 

Table of values of e-^ 

Table of common logarithms of e^ and e-^ 

Five-place table of natural logarithms 

Table of logarithms of T {x) . 

Three-place table of natural trigonometric functions 

Four-place table of common logarithms of numbers 

Four-place table of trigonometric functions 

Tables for reducing radians to degrees 

152 



73-80 

81-83 

84-87 

88-96 

97-106 

106-109 

109 

110-114 

115 

116-120 

121-123 

124-127 

127 

128, 129 

130-139 

140 

141 

142-145 

146-150 

151 



14 DAY USE 

RETURN TO DESK FROM WHICH BORROWED 

LOAN DEPT. 

This book is due on the last date stamped below, or 

on the date to which renewed. 

Renewed books are subject to immediate recall. 



NOV 8-1966 6 3 



4ftHc3c:W^ 



..Mr^ ' Bb-^ 



^ 



\J'u^' 



L.<;^'/^iH 



ij^fur 



M^^m^ 



AUG 21 1984 



CIRCULAT'^M nFPT. 



LD 21A-60ni-7,'66 
(G4427sl0)476B 



General Library 

University of California 

Berkeley 



